(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

This is #22 of Chapter 2 in Rudin's Real and Complex Analysis 3rd Ed.

Suppose that X is a metric space, with metric d, and that f : X -> [0,infinity] is lower semicontinuous, f(p) < infinity for at least one p in X. For n = 1,2,3,..., x in X, define

[tex]g_n(x) = \inf\,\{\,f(p) + nd(x,p)\,:\,p \in X\,\}[/tex]

and prove that

(i) |g_{n}(x) - g_{n}(y)| <= nd(x,y),

(ii) 0 <= g_{1}<= g_{2}<= ... <= f,

(iii) g_{n}(x) -> f(x) as n -> infinity, for all x in X.

2. Relevant equations

f is lower semicontinuous if for every real a, f^{-1}((a,infinity]) is open in X.

3. The attempt at a solution

This was a from a homework problem due in class earlier today, that just killed me to think about. (i) and (ii) were straightforward, and then I just could not figure out how to show that sup g_{n}= f. Because g_{n}is defined as an inf, what is stopping each subsequent g_{n}from increasing very very little? Perhaps the lower semicontinuity of f, which wasn't necessary for (i) or (ii)? Any help would be appreciated.

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# Homework Help: Rudin Semicontinuity Exercise

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