Uniqueness of Geometric Memorylessness

Part (a): Let $N$ be a positive-integer-valued random variable satisfying $P(N > m + n \mid N > m) = P(N > n)$ for all $m, n \in \mathbb{Z}_{\geq 0}$. Prove that $N$ must follow a geometric distribution.

Part (b): For $N \sim \operatorname{Geom}(p)$, compute $E[N^2 \mid N > k]$ using memorylessness and verify that $\operatorname{Var}(N \mid N > k) = \operatorname{Var}(N)$.