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## Homework Statement

Show that an increasing sequence is convergent if it has a convergent subsequence.

## The Attempt at a Solution

Suppose x

_{jn}is a subsequence of x

_{n}and x

_{jn}→x.

Therefore [itex]\exists[/itex]N such that j

_{n}>N implies |x

_{jn}-x|<[itex]\epsilon[/itex]

It follows that n>j

_{n}>N implies |x

_{n}-x|<[itex]\epsilon[/itex]

Therefore x

_{n}→x

The solution that I've been given is much more complicated I'm just wondering whether my simpler solution is also correct.