real analysis - Is there any analogue of Bolzano-Weierstrass theorem for sequence of functions?

Bolzano-Weierstrass theorem states that every bounded sequence has a convergent subsequence.
Now suppose ${f_n}$ is a sequence of functions on a given domain $D$.Does there exist a similar result involving subsequences as in Bolzano-Weierstrass theorem?
I tried to give a result:

If the sequence of functions is such that $\exists M \in\Bbb R$ satisfying $|f_n(x)|\leq M\forall x\in D, \forall n\in \Bbb N$,then does there exist a pointwise convergent subsequence ${f_{r_n}}$ on the domain ?

Then I got a counterexample:
Suppose domain is $\Bbb R$.Construct ${f_n}$ as follows:
$f_n(x)=sin(nx)$ which is bounded by $1$ but I think no subsequence of it is pointwise convergent.
So,Can anyone suggest me a similar result as in Bolzano-Weierstrass theorem for a sequence of functions?

Answer

The Arzela-Ascoli theorem states: consider a family of real-valued functions $\mathcal{F} = \{f_n \vert n \in \mathbb{N}\}$ defined on a compact set. If $\mathcal{F}$ is uniformly bounded and equicontinuous, then it is precompact.

To answer your question, if a given sequence $(f_n)$ meets the boundedness and equicontinuity conditions, then there exists a uniformly convergent subsequence $(f_{n_k})$.

There are other, equivalent statements of the theorem, but I believe this will be the most useful for your purposes.