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Could you help me prove the following?

Let S be the set of function that are the pointwise limit of continuous functions, $\{h _n\} \subset S$ with max$_{x \in [0,1]} |h_n(x)|< A_n$ and $\sum A_n < \infty$. Then $\sum h_n \in S$.

I would need something like this to finish solving a problem concerning continuous functions convergent to an increasing function.

Thanks

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Each $h_n$ can be approximated pointwise by a sequence $(h_{n,k},k\geqslant 1)$ of continuous functions. We can assume without loss of generality that $|h_{n,k}(x)|\leqslant A_n$ for all $k$ (if it's not the case we truncate).

Let $g_k:=\sum_{j=1}^kh_{k,j}$, a continuous function. We shall see that $g_k\to h:=\sum_nh_n$ pointwise. Fix $x\in [0,1]$ and $\varepsilon>0$. Fix $N$ such that $\sum_{j\geqslant N}A_j<\varepsilon$. Consider an integer $k\geqslant N$. Then $$|g_k(x)-h(x)|\leqslant \sum_{j\geqslant k+1}A_j+\sum_{j\geqslant N}A_j+\sum_{j=1}^N|h_{n,k}(x)-h_j(x)|,$$ hence $$\limsup_{k\to +\infty}|g_k(x)-h(x)|\leqslant 2\varepsilon.$$

Functions which can be approximated pointwise by continuous functions are called Baire's class one functions (it can be helpful to know that for further properties).

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  • $\begingroup$ Thanks. Do you think you could help me prove that characteristic functions of intervals also belong to this set S? $\endgroup$ – Don Apr 7 '13 at 10:27
  • $\begingroup$ As it's a different question, you could ask a new question. $\endgroup$ – Davide Giraudo Apr 7 '13 at 10:29
  • $\begingroup$ Ok, I will do that :) Here it is: math.stackexchange.com/questions/353756/… $\endgroup$ – Don Apr 7 '13 at 10:33

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