Re: simple analysis exercise (uniform continuity)

On 24-08-2007 17:39, Kiuhnm wrote:

Now, take _x_ and _y_ in cl(S) such that d(x,y) < d. Take x',y' in S
such that d(x,x'),d(y,y') < d. Deduce that

d(g(x),g(x')),d(g(y),g(y')) <= e/3.

Every time I can't solve a problem, it turns out that I have trouble with the very key point. At least I'm consistent :-)

I /know/ that d(g(x),g(x')) <= e/3 (and that '<' would be wrong) but I can't find a simple way to prove it.

Oh! *That* problem! :-)

Well, _x_ is in the closure of S. Pick some sequence (x_n)_n of elements
of S such that lim_n x_n = x. Then, since d(x,x') < d, d(x_n,x') < d, if
_n_ is large enough. For those values of _n_, d(g(x_n),g(x')) < e/3, and
therefore:

d(g(x),g(x')) = d(lim_n g(x_n),g(x')) = lim_n d(g(x_n),g(x')) <= e/3.

Best regards,

Jose Carlos Santos
.

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