I had an exam on functional analysis some time ago, and one of the questions I couldn't make any sense out was the following:

Let $\Omega\subset \mathbb{R}$ and $\{f\_n\}$ a sequence of continuous functions from $\Omega$ to $\mathbb{R}$. If the following criteria are obeyed:

1. $\exists M>0$ such that $||f\_n||\_{\infty}< M$ $\forall n\in\mathbb{N}$,
2. $\Omega$ is compact,
3. The sequence $\{f\_n\}$ is uniform equicontinuous.

then the theorem of Arzèla Ascoli states that the sequence $\{f\_n\}$ has a subsequence which converges in the $||.||\_{\infty}$norm to a continuous function.

Show that the theorem is not true by stating counter examples in the cases:

1. (1) and (2) are obeyed, but not (3),
2. (1) and (3) are obeyed, but not (2),
3. (2) and (3) are obeyed, but not (1).

I spend a lot of time thinking about this, but I couldn't think of any counter examples. For a non-compact subset of $\mathbb{R}$ I tried $(0,1)$ (as it is not closed) and for a bounded sequence I was thinking of $f\_n = x^n$ but these didn't work. Can anyone help me with some counter examples and maybe a good way of thinking of them?