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In mathematics, a direct proof is a logical argument that demonstrates the truth of a proposition by a straightforward chain of reasoning based on the given assumptions and axioms. It involves no detours or assumptions contrary to the given statement. Here’s how it works:

• Identify the statement you want to prove.
• Use definitions, axioms, and previously established results to construct a proof.
• Ensure each logical step follows naturally from the preceding one.

When we apply a direct proof, like proving that every odd integer can be expressed as the difference of two squares, we proceed step-by-step, using basic algebra and a hint provided in the problem to establish this arithmetic fact.

The difference of squares is a fundamental algebraic identity used to simplify expressions and solve equations. It states that for any two numbers, say a and b, the difference of their squares can be expressed as:

\[ a^2 - b^2 = (a+b)(a-b) \]
Let's break it down:
• Start with two expressions, \( a^2 \) and \( b^2 \).
• Subtract \( b^2 \) from \( a^2 \).
• Factorize the result using the identity \( a^2 - b^2 \), which breaks down into the product of \( (a+b) \) and \( (a-b) \).

In the given proof, we use this identity to show that an odd integer can indeed be written as the difference of two squares. Specifically, the hint suggests substituting \( a = k + 1 \) and \( b = k \), where \( k \) is a positive integer. Doing the math, we find:

\[ (k+1)^2 - k^2 = 2k + 1 \]
Since \( 2k + 1 \) is always an odd number for any integer \( k \), this confirms our proof.

An odd integer is a number that cannot be evenly divided by 2, meaning it has a remainder of 1 when divided by 2. Odd integers have the general form:

\[ 2n + 1 \]
where \( n \) is any integer. Some examples of odd integers include:
• -3
• -1
• 1
• 3
• 5

This line of numbers continues indefinitely. In our proof, we rely on the property of odd numbers to show that they can indeed be represented as the difference of two squares. Specifically, if we can demonstrate that for every number of the form \( 2n + 1 \), there exist two numbers whose squares give the needed difference, this establishes the claim. By using variables and substituting into the difference of squares formula, we see the clear pattern that indeed supports our assertion. This caps off our direct proof logically and coherently.