Determining functions convergence

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Reference no: EM13118574

1. Show that if (f_n), (g_n) converges uniformly on the set A to f and g respectively,then (f_n+g_n) converges on A to f+g.

2. Show that if f_n(x) :=x+1/n and f(x) := x for x in the reals, then (f_n) converges uniformly on the reals to f, but the sequence ((f_n)^2) does not converge uniformly on the reals (thus the product of uniformly convergent sequences of functions may not converge uniformly).

3. Let (f_n) be a sequence of functions that converges uniformly to f on A and that satisfies |f_n(x)|< M for all n in the naturals and for all x in A. If g is continuous on the interval [-M,M], show that the sequence (g composed of f_n) (or g o f_n) converges uniformly to g composed of f (or g o f ) on A.

Reference no: EM13118574

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