Proving n is even ↔ n² is even


Definition: An integer n \in \mathbb{N} is even is there exists an integer k \in \mathbb{N} such that 2k = n.

Proof: Let n be an even integer. This implies there must exist an integer k such that 2k = n. To determine whether or not n^{2} is even, we can simply look at (2k)^{2} \leftrightarrow 4k^{2}. The integer 4k^{2} must now obviously be even since 2 \cdot 2k^2 = 4k^{2}. \blacksquare.

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