Rearrangement of a Sequence

Hi, I want to prove that a converging sequence will still converge to the same limit even if all the terms are permuted. I figured the problematic terms are the first $N-1$ terms since they can be located anywhere in the permuted sequence. To get rid of them, I want to find the largest permuted index of these finite terms and define that as the new starting point for the tail sequence. Here's what I got:

Suppose $\{a_n\}$ converges to $L$ . Let $\phi : \mathbb{N} \rightarrow \mathbb{N}$ be a bijection. Define a new sequence by $\{ a_{\phi(n)} \}$ . We want to show this new sequence converges to $L$ .

Given any $\epsilon > 0$ , there is some $N' \in \mathbb{N}$ such that if $n \geq N'$ , then $d(a_n, L) < \epsilon$ . Define $N = \max \{\phi(1), \phi(2), \ldots, \phi(N'), N' \}$ and suppose $\phi(n) \geq N$ . Then $d(a_{\phi(n)}, L) \leq \epsilon$ .

It seems like that should do it but I'm always skeptical when it seems easy. Thanks.