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## Homework Statement

Prove that every convergent sequence has a monotone subsequence.

## Homework Equations

## The Attempt at a Solution

Define ##L## to be the limit of ##(a_n)##. Then every ##\epsilon##-ball about L contains infinitely many points. Note that ##(L, \infty)## or ##(-\infty, L)## (or both) has infinitely many elements. Suppose that ##(L, \infty)## has infinitely many elements. Choose ##n_1## such that ##a_{n_1} \in (L, \infty)##. Choose ##n_2 \in (L,a_{n_1})##. In general, choose ##n_{k+1}## such that ##a_{n_{k+1}} \in (L, a_{n_{k}})##. This assignment can always be made since there are infinitely many elements of ##(a_n)## in ##(L, a_{n_{k}})## to choose from such that ##a_{n_{k+1}} \in (L, a_{n_{k}})##.