The best examples of 3 practical examples of direct proof (and more)

Starting with real examples of 3 practical examples of direct proof

Let’s begin with three of the best examples of direct proof that almost every instructor loves. These are the “training wheels” of proof writing: simple enough to follow, but rich enough to teach you the pattern you’ll reuse in harder problems.

Each example follows the same basic recipe:

• Assume the hypothesis is true.
• Use definitions and known facts.
• Step logically to the conclusion.

These are the core examples of 3 practical examples of direct proof that you should be able to write from memory.

Example 1: Sum of two even integers is even

Statement. For all integers \(m\) and \(n\), if \(m\) and \(n\) are even, then \(m + n\) is even.

This is the classic example of a direct proof, and it’s usually one of the first real examples students see.

Step-by-step direct proof:

We start by assuming the hypothesis: let \(m\) and \(n\) be even integers. By definition of even, there exist integers \(k\) and \(\ell\) such that
[
m = 2k \quad \text{and} \quad n = 2\ell.
]
Now add them:
[
m + n = 2k + 2\ell = 2(k + \ell).
]
Since \(k + \ell\) is an integer (the sum of two integers is an integer), we can call it \(p\). Then
[
m + n = 2p,
]
so \(m + n\) is even by definition.

That’s it. You assumed the “if” part, used the definition of even, and arrived at the “then” part. This is a textbook example of 3 practical examples of direct proof because it’s short, transparent, and completely algebraic.

Example 2: Product of two odd integers is odd

Statement. For all integers \(a\) and \(b\), if \(a\) and \(b\) are odd, then \(ab\) is odd.

This is a natural partner to the first example, and many instructors treat these as two of the best examples to test whether you really understand how direct proof works.

Proof:

Assume \(a\) and \(b\) are odd integers. By definition of odd, there exist integers \(r\) and \(s\) such that
[
a = 2r + 1 \quad \text{and} \quad b = 2s + 1.
]
Now multiply:
[
ab = (2r + 1)(2s + 1) = 4rs + 2r + 2s + 1.
]
Factor out a 2 from the first three terms:
[
ab = 2(2rs + r + s) + 1.
]
Let \(t = 2rs + r + s\), which is an integer. Then
[
ab = 2t + 1,
]
so \(ab\) is odd.

Again, this is a clean example of a direct proof: no tricks, no indirect reasoning, just definitions and algebra.

Example 3: If n is even, then n² is even

Statement. For every integer \(n\), if \(n\) is even, then \(n^2\) is even.

This is the third member in our set of examples of 3 practical examples of direct proof. It’s a little different from the first two, but it uses exactly the same structure.

Proof:

Assume \(n\) is an even integer. Then there exists an integer \(k\) such that
[
n = 2k.
]
Square both sides:
[
n^2 = (2k)^2 = 4k^2 = 2(2k^2).
]
Let \(q = 2k^2\), which is an integer. Then
[
n^2 = 2q,
]
so \(n^2\) is even by definition.

These three are the standard “starter set” when instructors talk about examples of 3 practical examples of direct proof in textbooks and introductory proof courses.

Going beyond: more real examples of direct proof in action

Once you’re comfortable with the first three, you can stretch the same pattern to more interesting statements. Below are several more real examples of direct proof that you’re likely to meet in discrete math, number theory, or early analysis.

Example 4: Sum of two rational numbers is rational

Statement. For all rational numbers \(x\) and \(y\), the sum \(x + y\) is rational.

Proof:

Assume \(x\) and \(y\) are rational. By definition, there exist integers \(a, b, c, d\) with \(b \neq 0\) and \(d \neq 0\) such that
[
x = \frac{a}{b}, \quad y = \frac{c}{d}.
]
Then
[
x + y = \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}.
]
The numerator \(ad + bc\) is an integer (integers are closed under multiplication and addition), and the denominator \(bd\) is a nonzero integer (product of nonzero integers is nonzero). So \(x + y\) is a ratio of two integers with nonzero denominator, which means \(x + y\) is rational.

This is one of the best examples to show how direct proof works beyond simple parity arguments.

Example 5: If n is divisible by 6, then n is divisible by 3

Statement. Let \(n\) be an integer. If 6 divides \(n\), then 3 divides \(n\).

Proof:

Assume 6 divides \(n\). By definition of divisibility, there exists an integer \(k\) such that
[
n = 6k.
]
Rewrite 6 as \(3 \cdot 2\):
[
n = 3(2k).
]
Since \(2k\) is an integer, call it \(m\). Then \(n = 3m\), so 3 divides \(n\).

This kind of direct proof appears constantly in number theory and is a realistic example of how you’d reason about factors in competition problems or coding interviews.

Example 6: If 0 < a < b, then a² < b² (for positive reals)

Statement. Let \(a\) and \(b\) be real numbers. If \(0 < a < b\), then \(a^2 < b^2\).

Proof:

Assume \(0 < a < b\). Then \(b - a > 0\), and \(b + a > 0\) because \(b > a > 0\). Now compute
[
b^2 - a^2 = (b - a)(b + a).
]
Both factors are positive, so their product is positive:
[
b^2 - a^2 > 0.
]
Therefore \(b^2 > a^2\), which is equivalent to \(a^2 < b^2\).

This example of direct proof is closer to what you’ll see in calculus and real analysis, but the structure is the same: assume the hypothesis, push algebra, reach the conclusion.

For a deeper treatment of inequalities and real numbers, you can look at open course materials from universities such as MIT OpenCourseWare or Harvard’s math resources.

Example 7: If a and b are positive, then a + b > a

Statement. Let \(a\) and \(b\) be real numbers. If \(a > 0\) and \(b > 0\), then \(a + b > a\).

Proof:

Assume \(a > 0\) and \(b > 0\). Then
[
a + b > a + 0 = a,
]
because adding a positive number \(b\) to \(a\) increases its value. Formally, from \(b > 0\) we know that adding \(b\) to both sides of \(a > 0\) preserves the inequality direction. So \(a + b > a\).

This is almost embarrassingly simple, but it’s a good reminder that direct proofs don’t have to be fancy. In applied fields like statistics or economics, arguments about inequalities often quietly rely on this kind of direct reasoning.

Example 8: If a sequence is increasing, then each term is at least the first term

This one shows up in early analysis and data science settings when people talk about monotone sequences.

Statement. Let \((x_n)\) be a sequence of real numbers such that for all integers \(n \ge 1\), \(x_{n+1} \ge x_n\). Then for all \(n \ge 1\), \(x_n \ge x_1\).

Proof (direct, by simple induction flavor):

We want to show: for any integer \(n \ge 1\), \(x_n \ge x_1\).

Take an arbitrary integer \(n \ge 1\). From the hypothesis, we know
[
x_2 \ge x_1, \quad x_3 \ge x_2, \quad x_4 \ge x_3, \; \dots, \; x_n \ge x_{n-1}.
]
Chaining these inequalities gives
[
x_n \ge x_{n-1} \ge x_{n-2} \ge \dots \ge x_2 \ge x_1. ] Therefore \(x_n \ge x_1\).

This is still a direct proof: we assume the condition “the sequence is increasing” and use it directly, step by step, to get the desired inequality.

For more formal treatments of sequences and limits, you can check undergraduate analysis notes from universities like UCLA or Princeton.

How to recognize problems suited for direct proof

At this point, we’ve gone far beyond just three. We’ve seen examples of 3 practical examples of direct proof and then several more that appear in real coursework. So how do you know when to use a direct proof on a new problem?

Look for statements that:

• Have a clear “if–then” structure: “If P, then Q.”
• Use definitions you already know well (even, odd, rational, divisible, increasing, positive).
• Don’t involve negations or contradictions in the wording.

For example, “If a number is divisible by 4, then it is even” almost begs for a direct proof using the definition of divisibility, just like our example with 6 and 3. On the other hand, statements that say “There is no integer with property X” often invite proof by contradiction or contrapositive instead.

Many undergraduate proof courses and discrete math classes, such as those described in materials from Carnegie Mellon University, start with exactly these kinds of examples. They’re not just toy problems; they train you to unpack definitions and move line by line from assumptions to conclusions.

Why these are the best examples to practice with in 2024–2025

In 2024–2025, more math programs are shifting to proof-based courses earlier, especially for students headed into computer science, data science, and quantitative finance. That means you’ll see direct proof not just in pure math, but in:

• Algorithm correctness arguments (e.g., proving that a loop invariant holds at each step).
• Data structure properties (e.g., proving that a binary search tree with certain rules always keeps keys ordered).
• Probability and statistics (e.g., proving linearity of expectation for random variables).

The examples of 3 practical examples of direct proof we walked through map surprisingly well to these areas. The habit you build—assume the setup, use definitions, and step logically to the claim—is exactly the habit you’ll need when you explain why a sorting algorithm really does output a sorted list, or why a Markov chain has certain properties.

If you want to see how rigorous reasoning plays out in applied contexts, sites like NIST.gov and NSF.gov often publish technical reports that rely heavily on clear, direct logical arguments, even when they’re not labeled as “direct proofs.”

FAQ: common questions about examples of direct proof

Q1. Can you give another simple example of direct proof with integers?
Yes. Here’s a quick one: “If \(n\) is an integer and \(n\) is odd, then \(n^2\) is odd.” You assume \(n = 2k + 1\), expand \(n^2 = (2k + 1)^2\), simplify to \(2m + 1\) form, and you’re done. It follows exactly the pattern of our product-of-odds example.

Q2. Are examples of 3 practical examples of direct proof enough to master the technique?
They’re a good starting point, but not enough on their own. You should practice with a wider range of problems—divisibility, inequalities, rational/irrational numbers, and simple sequence properties. The more varied your set of examples of direct proof, the easier it is to recognize when a new problem can be attacked directly.

Q3. How do I know if my argument is a valid example of direct proof?
Check three things: you clearly assumed the hypothesis, you only used correct logical steps and known definitions, and you ended exactly at the statement you were supposed to prove. If you never flipped the statement to its contrapositive and never argued by contradiction, you most likely have a direct proof.

Q4. Where can I find more examples of direct proof with solutions?
Many university course pages host free notes and problem sets. Look for “introduction to proofs” or “discrete mathematics” at .edu sites. For instance, MIT, Harvard, and other schools publish open materials that include worked examples, exercises, and solution outlines.

Q5. What is one real-world style example of direct proof from computer science?
A classic example is proving that a simple function that adds 1 to a counter variable and stops when it reaches \(n\) will always terminate and output \(n\). You assume the starting conditions, track the variable step by step, and show that after exactly \(n\) iterations, the loop stops with the correct value. It’s the same pattern as our number-theory examples, just written in the language of algorithms.

You don’t need dozens of fancy tricks to get comfortable with proofs. A small set of well-understood examples of 3 practical examples of direct proof—plus a handful of extra problems like the ones above—will give you the confidence to tackle much harder theorems later. Start with these, rewrite them in your own words, and then try creating similar statements to prove on your own. That’s where the real learning happens.