# Does uniform boundedness and monotonicity of a sequence of functions imply convergence

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?

• 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. Dec 15 '19 at 21:59

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