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Corollary of the M-test

If the sum of supremum norms is finite, then the corresponding series of functions converges uniformly.

Corollary of the M-test. Let EE be a set and let fn:E→Rf_n:E\to\mathbb R or fn:E→Cf_n:E\to\mathbb C be bounded functions. If

∑n=1∞∥fn∥∞\sum_{n=1}^\infty \|f_n\|_\infty

converges, where ∥fn∥∞=sup⁡x∈E∣fn(x)∣\|f_n\|_\infty=\sup_{x\in E}|f_n(x)|, then ∑n=1∞fn\sum_{n=1}^\infty f_n converges on EE and absolutely at each point.

Remarks

This is the with Mn=∥fn∥∞M_n=\|f_n\|_\infty.