Diverse examples of existential proofs in modern mathematics

Starting with concrete examples of existential proofs

Instead of beginning with abstract definitions, let’s start with real examples. The best examples of existential proofs share a common flavor: you walk away knowing something is out there, even if you never see it directly.

Here are several classic and modern situations where the conclusion is purely existential:

• There exist irrational numbers \(a, b\) such that \(a^b\) is rational.
• There exists a prime between any integer \(n > 1\) and \(2n\).
• There exists a graph with very high edge density but no large complete subgraph.
• There exists a continuous function that is nowhere differentiable.
• There exists a Turing machine that halts on some inputs but for which no algorithm can decide all halting cases.

Each of these is proved using a different style of existential argument. Taken together, they form some of the best examples of diverse examples of existential proofs that you can use to sharpen your proof strategy toolkit.

Classic number theory: examples of diverse examples of existential proofs

Number theory is a goldmine for existential reasoning. Many famous theorems state that certain numbers must exist, yet give you no simple formula for them.

Example of existence: infinitely many primes

Euclid’s proof that there are infinitely many primes is often presented as a proof by contradiction, but it is also an existential proof on repeat: for any finite list of primes, there exists another prime not on that list.

Sketch:

Assume you have a finite list of primes \(p_1, p_2, \dots, p_k\). Consider [ N = p_1 p_2 \cdots p_k + 1.
]
Any prime divisor \(q\) of \(N\) is not equal to any \(p_i\), so a new prime exists beyond your list. This is a clean example of an existential proof that, for every finite set of primes, there exists a new one.

Bertrand’s Postulate: a prime between \(n\) and \(2n\)

Bertrand’s Postulate (proved by Chebyshev in the 19th century) states:

For every integer \(n > 1\), there exists a prime \(p\) such that \(n < p < 2n\).

This is a stronger, more quantitative existential statement than Euclid’s. The standard proofs do not tell you which prime it is for a given \(n\), they only guarantee that such a prime exists.

Modern refinements use analytic number theory and ideas connected to the prime number theorem. For an accessible introduction to this style of reasoning, see the MIT OpenCourseWare number theory notes.

Irrational powers giving rational results

This is a classic classroom favorite among diverse examples of existential proofs in real analysis and algebra.

Claim: There exist irrational numbers \(a\) and \(b\) such that \(a^b\) is rational.

Proof outline:

Consider the number \(\sqrt{2}^{\sqrt{2}}\). Either this number is rational or irrational.

• If \(\sqrt{2}^{\sqrt{2}}\) is rational, then take \(a = \sqrt{2}\), \(b = \sqrt{2}\), and we are done.
• If \(\sqrt{2}^{\sqrt{2}}\) is irrational, then consider
  [
  a = \sqrt{2}^{\sqrt{2}}, \quad b = \sqrt{2}.
  ]
  Then
  [
  a^b = \left(\sqrt{2}^{\sqrt{2}}\right)^{\sqrt{2}} = \sqrt{2}^{2} = 2,
  ]
  which is rational.

Either way, there exists an irrational pair \((a, b)\) with \(a^b\) rational. This is a non-constructive example of an existential proof because we never actually determine whether \(\sqrt{2}^{\sqrt{2}}\) is rational or irrational; we argue by cases that some pair must work.

Analysis: real examples of wild continuous functions

Real analysis offers some of the most surprising examples of diverse examples of existential proofs. You learn early that continuous functions are “nice"—but existence theorems show they can be much stranger than your intuition suggests.

A continuous, nowhere differentiable function

The classic Weierstrass function is continuous everywhere and differentiable nowhere. The theorem can be phrased as:

There exists a function \(f : \mathbb{R} \to \mathbb{R}\) that is continuous at every real number but differentiable at no real number.

The original construction by Weierstrass is explicit, but many modern proofs in functional analysis show the existence of such functions using more abstract arguments, for example by showing that the set of nowhere differentiable continuous functions is dense in the space of continuous functions. Those proofs are existential in flavor: they show that “most” continuous functions have this wild property, even if you never write one down.

For a rigorous treatment, see functional analysis lecture notes from universities like Harvard or MIT, which often include these existence results.

Intermediate Value Theorem as an existential statement

The Intermediate Value Theorem (IVT) is usually presented as a continuity result, but it is also a very natural example of an existential proof:

If \(f\) is continuous on \([a, b]\) and \(f(a) < 0 < f(b)\), then there exists \(c \in (a, b)\) such that \(f(c) = 0\).

The standard \(\varepsilon\)–\(\delta\) or bisection-based proof does not tell you the exact value of \(c\); it only shows that some such \(c\) must exist. Numerical analysis then provides algorithms that approximate \(c\), but the underlying theorem is existential.

Graph theory and combinatorics: examples include probabilistic proofs

If you want modern, 2024-friendly examples of diverse examples of existential proofs, you head straight to probabilistic combinatorics. This is where the probabilistic method, pioneered by Paul Erdős, dominates.

Probabilistic method: graphs with surprising properties

A classic statement:

For sufficiently large \(n\), there exists a graph on \(n\) vertices with no large complete subgraph (no large clique) and yet with many edges.

Erdős showed that by taking a random graph and computing the probability that it has certain properties, you can prove that graphs with those properties must exist—even if you never explicitly construct one.

This is a textbook example of an existential proof: you prove that the set of “good” graphs has positive probability, so at least one such graph exists. Modern research in extremal graph theory and Ramsey theory still uses this style of argument extensively in 2024–2025.

For a modern reference, the probabilistic method is covered in graduate-level courses and notes available from institutions like Princeton University and other .edu sites.

Ramsey theory: large structured subsets must exist

Ramsey theory is built on existential statements of the form:

Given enough elements in a structure, there exists a large subset with a particular pattern.

For example, a classic theorem says that for any coloring of the edges of a sufficiently large complete graph in two colors, there exists a monochromatic complete subgraph of a specified size.

The proofs often combine counting arguments and the pigeonhole principle, yielding real examples of existential proofs that are extremely non-constructive. In many cases we know that certain Ramsey numbers exist but still do not know their exact values, even with current 2024 computational power.

Logic, computability, and 2024-era algorithmic examples

Modern theoretical computer science provides some of the most interesting 21st-century examples of diverse examples of existential proofs. Here, existential proofs often show that certain algorithms or objects must exist, even though we cannot efficiently find them.

Halting problem: existence of undecidable instances

Turing’s result on the halting problem can be viewed as an existential theorem:

There exists a Turing machine \(M\) such that no algorithm can decide, for all inputs, whether \(M\) halts on that input.

The proof constructs such an \(M\) indirectly, by assuming a universal halting-decider exists and then deriving a contradiction. This is a classic indirect existential proof: it identifies an object with a specific property (undecidability) without ever listing all its behaviors in a way an algorithm could fully analyze.

Complexity theory and hard instances (2024 context)

In complexity theory, statements like “P ≠ NP” (still unproved as of 2025) would have strong existential consequences. Even without that resolution, we already have theorems that look like this:

For every polynomial-time algorithm \(A\) for a certain NP-hard problem, there exists an input instance on which \(A\) performs poorly.

These are existential proofs about hard instances: they guarantee that some inputs force bad behavior. Recent work in average-case complexity and fine-grained complexity, presented at conferences like STOC and FOCS through 2024, continues to rely heavily on existential arguments of this flavor.

You can find introductory explanations of these ideas in lecture notes hosted on .edu domains, for example from Cornell University’s CS theory courses.

Real examples from modern probability and statistics

Probabilistic existence theorems are not limited to pure math. They show up in modern statistics and randomized algorithms, which are very much alive in 2024–2025 research.

Randomized algorithms: existence of good hash functions

In algorithm design, one often proves statements like:

There exists a hash function in a given family that yields few collisions on a particular dataset.

The proof usually goes like this: choose a hash function at random from a carefully designed family and compute the expected number of collisions. If the expected number is small, then there must exist at least one hash function in the family with few collisions. This is yet another clean example of an existential proof via expectation.

This style of argument underpins modern data structures such as universal hashing and is discussed in algorithm textbooks used widely in the US and internationally.

High-dimensional geometry and machine learning

In high-dimensional statistics and machine learning theory, many theorems assert the existence of classifiers, embeddings, or feature maps with certain generalization or separation properties. For example, using concentration inequalities, one can prove statements like:

For a random projection into a lower-dimensional space, there exists a choice of projection that approximately preserves all pairwise distances in a dataset.

This is related to the Johnson–Lindenstrauss lemma, whose standard proofs are existential and probabilistic. You show that a random projection works with high probability, so at least one such projection exists. These ideas continue to be active in 2024–2025 research on dimensionality reduction and privacy-preserving data analysis.

How to recognize and construct diverse examples of existential proofs

After seeing these real examples, it’s worth distilling what makes them tick. When you are trying to produce your own examples of diverse examples of existential proofs, look for these patterns:

• Quantifiers in the statement: Phrases like “there exists,” “for some,” or “at least one” signal that an existential proof is appropriate.
• Indirect construction: Many of the best examples use contradiction, counting, or probability to guarantee an object exists without naming it.
• Use of randomness: If you see a proof that starts with “pick an object uniformly at random,” you are likely looking at a probabilistic existential argument.
• Asymptotic or “for sufficiently large” language: Statements about behavior for large \(n\) often rely on existential proofs to show that certain structures appear once \(n\) crosses a threshold.

In practice, when you write your own proofs in courses or research, you will often:

• Assume no such object exists and derive a contradiction.
• Count how many objects are “bad” and show there must be at least one “good” one left.
• Use expectation or probability to argue that some configuration has the desired property.

These strategies mirror the real examples of existential proofs we’ve walked through: primes in intervals, wild functions in analysis, graphs with extreme properties, undecidable machines, and high-performing hash functions.

FAQ: common questions about examples of existential proofs

Q1. Can you give a simple example of an existential proof for beginners?
Yes. A very simple example of an existential proof is: “There exists an even prime number.” The proof is just: 2 is prime and even, so such a number exists. This is trivial, but it illustrates the pattern: identify one object with the property in question.

Q2. Are all existential proofs non-constructive?
No. Some examples of existential proofs are constructive: you actually build or exhibit the object. Others, like many probabilistic or contradiction-based arguments, are non-constructive: they guarantee existence without showing you a concrete example. The irrational power example and many Ramsey theory results are standard non-constructive cases.

Q3. Why are probabilistic methods so common in modern existential proofs?
Because they are powerful and flexible. By analyzing random objects, you can often show that the probability of success is positive, which immediately implies that some object with the desired property exists. This has become a standard approach in combinatorics, computer science, and high-dimensional statistics, especially in research through 2024–2025.

Q4. How do existential proofs show up in real-world applications?
Real examples include guarantees about cryptographic keys, hash functions with low collision rates, and worst-case instances for algorithms. In each case, the theory may only assert that such objects exist, while practical implementations use heuristics or randomized search to find them.

Q5. Where can I study more examples of diverse examples of existential proofs?
Advanced undergraduate and graduate courses in real analysis, number theory, graph theory, and theoretical computer science are full of these proofs. Look for open course materials from major universities (.edu domains), and pay attention whenever you see the phrase “there exists” in theorem statements—those are your best examples to study.