Confused by the Bolzano Weierstrass Theorem

It looks like you're quite on top of the first lemma you describe. Please clarify if you need any help with that.

For the second one, I think your confusion is caused by you misunderstanding "bounded", or at least getting the wrong mental picture when you see it. A bounded sequence still contains infinitely many points -- that it is bounded means that all of the points lie within the same interval on the $y$-axis.

One example is the sequence $a_n = \sin(n)$ for $0\le n\lt \infty$. Here $n$ can be as large as you please, but every $a_n$ is between $-1$ and $1$, which makes the sequence bounded.

Therefore, Bolzano-Weierstrass says that $(\sin(n))_n$ contains a convergent subsequence. It is not easy to write down one explicitly, though. (Since we're mathematicians here, $\sin(n)$ treats $n$ as an angle in radians, so there's no value in the sequence that repeats exactly).

If $l$ is the supremum of your sequence $(a_n)$ then, by the very definition of the supremum we have the following:

For every $\varepsilon >0$ there is some $n$ such that $a_n\geq l-\varepsilon$.

But since your sequence is monotonically increasing, you have $a_m \geq l-\varepsilon$ for every $m\geq n$. This is exactly what it means for the sequence $(a_n)$ to converge to $l$.