Uniform continuity, cauchy sequences

missavvy

1. The problem statement, all variables and given/known data
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.


2. Relevant equations



3. 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?

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).