Suppose {f n } and {g n } are bounded sequences each of which converges uniforml

Question

Suppose {fn} and {gn} are bounded sequences each of which converges uniformly on a set A in a metric space S to functions f and g, respectively. Show that the sequence {fngn} converges uniformly to fg on A.

Explanation / Answer

f_n --> f = > || f_n - f || --> 0 as n --> infinity , f_n are bounded => || f_n || and ||f|| < = K > 0 g_n --> g => || g_n - g || --> 0 as n --> infinity , g_n are bounded => || g_n || and ||f|| < = L > 0 =>|| f_n g_n - f g|| = ||f_n g_n - f g_n + fg_n - fg|| < ||f_n g_n - f g_n|| + ||f g_n - fg|| [by triangle inequality.] < ||g_n|| || f_n - f || + ||f|| ||g_n - g|| infinity, => lhs goes to zero as n--> infinity So f_ng_n --> fg as n --> infinity. have a nice day !!

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