- #1

gotmilk04

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## Homework Statement

Show that if (x[itex]_{n}[/itex]) is a sequence in a metric space (E,d) which converges to some x[itex]\in[/itex]E, then (f(x[itex]_{n}[/itex])) is a convergent sequence in the reals (for its usual metric).

## Homework Equations

Since (x[itex]_{n}[/itex]) converges to x, for all ε>0, there exists N such that for all n[itex]\geq[/itex]N, d(x[itex]_{n}[/itex],x)<ε.

So |x-x[itex]_{n}[/itex]|<ε

## The Attempt at a Solution

I understand that this will prove continuity, but I'm not sure how to get from d(x[itex]_{n}[/itex],x)<ε to what we want: d(f(x[itex]_{n}[/itex]_,f(x))<ε