7/23/2019 33038

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proof of Dinis theorem

mathcam

2013-03-21 14:17:21

Without loss of generality we will assume that Xis compact and, by replac-ing fn with f fn, that the net converges monotonically to 0.

Let > 0. For eachx X, we can choose an nx, such that fnx(x) < /2.Since fnx is continuous, there is an open neighbourhood Ux of x, such that

for each y Ux, we have fnx(y) < /2. The open setsUx cover X, which iscompact, so we can choose finitely manyx1, . . . , xk such that the Uxi also coverX. Then, ifN nx1 , . . . , nxk , we have fn(x)< for each n N and x X,since the sequence fn is monotonically decreasing. Thus, {fn} converges to 0uniformly onX, which was to be proven.

ProofOfDinisTheorem created: 2013-03-21 by: mathcam version: 33038Privacysetting: 1 Proof 54A20

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