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Uniform continuity, cauchy sequences

  • Thread starter missavvy
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  • #1
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Homework Statement


If f:S->Rm is uniformly continuous on S, and {xk} is Cauchy in S show that {f(xk)} 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, {xn} and {xp}

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

Then using the fact that f is uniformly continuous, |f(xn)-f(xp)| < [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?
 

Answers and Replies

  • #2
radou
Homework Helper
3,115
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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).
 

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