The proposition and part of its proof are given below:
I am sorry for this question if it is very easy to answer:
In the first line of the proof why the uniform convergence and each $f_{n}$ being bounded leads to the limit function, $f$ is bounded?
2026-04-03 21:18:27.1775251107
Why the limit function is bounded?
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$|f-f_N| <1$ for some large $N$ by uniform convergence, and then $|f| \leq |f-f_N|+|f_N|$, so $f$ is bounded.