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jdcasey9
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Homework Statement
Prove that f:(M,d) -> (N,p) is uniformly continuous if and only if p(f(xn), f(yn)) -> 0 for any pair of sequences (xn) and (yn) in M satisfying d(xn, yn) -> 0.
Homework Equations
The Attempt at a Solution
First, let f:(M,d)->(N,p) be uniformly continuous.
Let [tex]\epsilon[/tex]=2[tex]\delta[/tex].
lf(xn)-f(yn)l [tex]\leq[/tex] lf(xn)-xnl + lxn-f(yn)l [tex]\leq[/tex] lf(xn)-xnl + lxn-ynl + lyn-f(yn)l < [tex]\delta[/tex] + 0 + [tex]\delta[/tex]= 2[tex]\delta[/tex] =[tex]\epsilon[/tex]
(because f is uniformly continuous)
Therefore, p(f(xn), f(yn))->0.
Second, let p(f(xn), f(yn)) -> 0 for (xn), (yn) in M such that d(xn, yn) ->0.
We can do this nearly the same way, except at the end we say:
lf(xn)-f(yn)l [tex]\leq[/tex] lf(xn)-xnl + lxn-ynl + lyn-f(yn)l -> 0 so it must be uniformly continuous.