What are the rules if you have a sequence [itex]f_n[/itex] of real-valued functions on [itex]\mathbb R[/itex] and consider the sequence [itex]f_n(x_n)[/itex], where [itex]x_n[/itex] is some sequence of real numbers that converges: [itex]x_n \to x[/itex]. All I have found is an exercise in Baby Rudin that says that if [itex]f_n \to f[/itex] uniformly on [itex]E[/itex], then [itex]f_n(x_n) \to f(x)[/itex] if [itex]x_n \to x[/itex] is in [itex]E[/itex]. But the exercise seems to indicate that it is possible to have [itex]f_n(x_n) \to f(x)[/itex] for every sequence [itex]x_n\to x[/itex] without having [itex]f_n \to f[/itex] uniformly. (I believe the canonical example [itex]f_n(x) = x^n[/itex] on [itex]E = [0,1][/itex] works here.)(adsbygoogle = window.adsbygoogle || []).push({});

I ask because I recently had a colleague who claimed that if [itex]x_n \to x[/itex], then [itex] \left( 1 + \frac{x}{n} \right)^n \to e^x[/itex]. She asked what the rule that made this possible was, and I replied that I wasn't sure if it was, in fact, true, since [itex]f_n(x) = \left( 1 + \frac xn \right)^n[/itex] obviously does not converge uniformly to [itex]e^x[/itex] on [itex]\mathbb R[/itex] (even though, obviously, [itex]f_n \to e^x[/itex] pointwise).

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# A sequence of functions evaluated at a sequence

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