Problem 8

Prove that every natural number is either even or odd.

For that we prove using induction that every natural number can be written as $2k$ (even) or $2k + 1$ (odd) for some $k \in Z$. Starting with $n = 1$ we get $2 \cdot 0 + 1 \in N$ for $k = 0$. Now we show that for an even $n$ we get an odd $n + 1$. $$n + 1 = 2k + 1$$ And vis versa (odd to even): $$n + 1 = 2k + 1 + 1 = 2k + 2 = 2(k + 1) = 2k^*$$


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