- #1
missavvy
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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?