# Questions tagged [uniform-convergence]

For sequences of functions, uniform convergence is a mode of convergence stronger than pointwise convergence, preserving certain properties such as continuity. This tag should be used with the tag [convergence].

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### Uniform convergence of polynomial approximation on Schwartz space

I have a question regarding uniform convergence of basis expansion in Schwartz space. For $L^2(\mathbb{R},\lambda)$, $\lambda$ Lebesgue measure, the partial sums of basis expansion (Hermite functions) ...
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### Is $S_n$ is uniformly convergent ? Yes/NO

Is $\displaystyle S_n = \sum_{n=1}^{\infty} 2^n \sin\frac{1}{3^nx}$ uniformly convergent on the interval $[1, \infty)$ ? True /false My attempt : NO, the given series will not uniformly ...
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### Uniform Cauchyness

Assume the sum of $|f_n|$ is uniformly cauchy. Does this imply that the sum of $(f_n)$ is uniformly cauchy? My reasoning is yes, since every $f_n$ is bounded by $|f_n|$.
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### If a sequence of functions $f_n$ converges uniformly to $f$, does this imply $f_n$ is continuous? How to prove?

I know I can find an $f_n$ that converges to f uniformly, but can we use this to argue $f_n$ is continuous? Is there a theorem I can state?
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