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Perhaps a dumb question, but suppose we know that a sequence of functions $\{f_n\}$ converges pointwise to $f$. Suppose we are then able to prove that $\{f_n\}$ converges uniformly (without any reference to $f$). Then does $\{f_n\}$ necessarily converge uniformly to $f$? Why couldn't it be some other function?

Thank you.

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If it converges uniformly then from the definition you have pointwise convergence and by the uniqueness of the limit the pointwise limit and uniform limit must be the same.

If you think about it the $n_\epsilon$ required for the uniform convergence works for any fixed $x$.

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The pointwise limit is unique for the exact same reason that the limit of any sequence of real numbers is unique (fix an $x$ and a sequence of real numbers is exactly what you have).

If you have shown that a sequence of functions converges uniformly to something, it is easy to see that it also converges pointwise to that something (think about what the order of quantifiers is saying). Now by the first fact, that something must in fact be $f$.

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The Cauchy criterion is satisfied uniformly for a uniformly convergent sequence of functions. You first show for any $\epsilon > 0$ there is an $N(\epsilon)$ such that for all $m >n >N$ and all $x$ we have

$$|f_m(x) - f_n(x)| < \epsilon$$

Now take the pointwise limit as $m \to \infty$ to get $|f_n(x) - f(x)| \leqslant \epsilon$ for all $n > N$ and each $x$ which verifies the limiting function.

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