Proving the Oddity of Squares: A Direct Proof Approach

Claim: The square of an odd number is odd.

Proof: First, observe that the statement the square of an odd number is odd is equivalent to an implication of the form: If n is odd then n². To prove this statement, let us assume that n is odd. That is n=2k+1 for some integer k. Next,

n² = (2k+1)²
 = 4k²+4k+1
 = 2(2k²+2k)+1

Letting ℓ=2k²+2k, it is clear that ℓ is an integer as well and consequently, n²=2ℓ+1 which is clearly odd as well.