Rudin Semicontinuity Exercise

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Homework Statement



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) |gn(x) - gn(y)| <= nd(x,y),
(ii) 0 <= g1 <= g2 <= ... <= f,
(iii) gn(x) -> f(x) as n -> infinity, for all x in X.

Homework Equations



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

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 gn = f. Because gn is defined as an inf, what is stopping each subsequent gn 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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