Unity Function is Completely Multiplicative
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Theorem
Let $f_1: \Z_{> 0} \to \Z_{> 0}$ be the constant function:
- $\forall n \in \Z_{> 0}: f_1 \left({n}\right) = 1$
Then $f_1$ is completely multiplicative.
Proof
- $\forall m, n \in \Z_{> 0}: f_1 \left({m n}\right) = 1 = f_1 \left({m}\right) f_1 \left({n}\right)$
$\blacksquare$