Question: Show that if fn(x) := x + 1/n and f(x) := x for x R, then (fn) converges uniformly on R to f, but

Show that if fn(x) := x + 1/n and f(x) := x for x ∈ R, then (fn) converges uniformly on R to f, but the sequence (f2n) does not converge uniformly on R. (Thus the product of uniformly convergent sequences of functions may not converge uniformly.)

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