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The monotone convergence theorem states for sequences of numbers that a monotonic and bounded sequence converges.

Is there an analog for a sequence of uniformly-bounded, monotonic for each $x$, functions $\{f_n(x)\}$? Does this imply uniform convergence, convergence, or nothing?

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  • $\begingroup$ Even if you require your functions to be continuous and monotone you can’t conclude anything, as the usual example ${x^n} n \in N$ with $x \in [0,1]$ shows. $\endgroup$
    – user622002
    Dec 15 '19 at 21:59
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Let $(f_n)_{n\in \mathbb{N}}$ be your sequence of real valued functions. For each $x$ in their domain, the sequence $(f_n(x))_{n\in \mathbb{N}}$ is monotone and bounded, so is convergent. It follows that the sequence $(f_n)_{n\in \mathbb{N}}$ converges pointwise to the function which maps each $x$ to the limit of $(f_n(x))_{n\in \mathbb{N}}$.

We can't conclude uniform convergence for example if $$f_n(x) = \left\{ \begin{array}{lr} 0 & \text{ if }0<x< \frac{1}{n}\\ 1 & \text{otherwise} \end{array}\right\}$$

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