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## Homework Statement

If f:S->R

^{m}is uniformly continuous on S, and {x

_{k}} is Cauchy in S show that {f(x

_{k})} is also cauchy.

## Homework Equations

## The Attempt at a Solution

Since f is uniformly continuous,

[tex]\forall[/tex][tex]\epsilon[/tex]>0, [tex]\exists[/tex][tex]\delta[/tex]>0: [tex]\forall[/tex]x, y ∈ S, |x-y| < [tex]\delta[/tex] => |f(x)-f(y)| < [tex]\epsilon[/tex]

So I said that let x, y be sequences, {x

_{n}} and {x

_{p}}

Since {xn} is Cauchy, [tex]\forall[/tex][tex]\epsilon[/tex]>0, [tex]\exists[/tex]N : [tex]\forall[/tex]n,p [tex]\geq[/tex] N , |x

_{n}-x

_{p}| < [tex]\epsilon[/tex]

Then using the fact that f is uniformly continuous, |f(x

_{n})-f(x

_{p})| < [tex]\epsilon[/tex]

I don't think this is right.. am I allowed to replace those x's and f(x)'s with the sequences for example?