"Let [tex]h[/tex] and [tex]g[/tex] be uniformly continuous on [tex]I\subset\mathbb{R}[/tex] and are both bounded. Show that [tex]hg[/tex] is uniformly continuous."(adsbygoogle = window.adsbygoogle || []).push({});

h and g are uniformly continuous and bounded on I implies that h and g are continuous and bounded on I. This implies that hg (h times g) is continuous and bounded on I. Let f(x) = h(x)g(x).

Assume that f is continuous on I but not uniformly, then there exists e>0 such that for all d>0, there exists x and y in I such that |x-y|<d and |f(y)-f(x)|>e. In particular, for every n>0, there exists x_n and y_n in I such that |y_n - x_n|<1/n and |f(y_n)-f(x_n)|>e. Since {x_n} is a sequence in I, by the Bolzano-Weierstrass theorem, there exists a subsequence x_n_p converging to z in I. As y_n_p = x_n_p + (y_n_p - x_n_p) for every n, and {y_n_p - x_n_p} converges to 0, {y_n_p} also converges to z. Since f is continuous at z, {f(y_n_p)-f(x_n_p)} converges to f(z)-f(z) = 0. This contradicts the assumption that |f(y_n_p)-f(x_n_p)|>e for all n. Hence, f must be uniformly continuous on I.

Does anyone see any flaws in that argument? I did it without referring to the fact that h and g are both bounded.

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# Homework Help: Product of Uniformly Continuous Functions

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