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 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 converges to . Let be a bijection. Define a new sequence by . We want to show this new sequence converges to .
Given any , there is some such that if , then . Define and suppose . Then .
It seems like that should do it but I'm always skeptical when it seems easy. Thanks.