Uniform continuity, cauchy sequences

[missavvy]

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,

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

So I said that let x, y be sequences, {x_n} and {x_p}

Since {xn} is Cauchy, $$\forall$$$$\epsilon$$>0, $$\exists$$N : $$\forall$$n,p $$\geq$$ N , |x_n-x_p| < $$\epsilon$$

Then using the fact that f is uniformly continuous, |f(x_n)-f(x_p)| < $$\epsilon$$

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

 

[radou]

Yes, basically, that's OK. You only need to be aware of the fact that for exactly this δ > 0 you found N (using the fact that (xn) is Cauchy).