I have a small doubt, the monotone convergence theorem says that for any increasing and convergent sequence, the limit of the sequence equal its supremum. But shouldn't this also be true for the more general statement, that for any increasing sequence, the limit of the sequence equal its supremum?
For example, if we have an unbounded increasing sequence, then the limit and the supremum should both be infinity, right?
If the sequence has a limit, then it is convergent and the monotone convergence theorem applies. If the sequence doesn't converge, then it doesn't have a limit, and so you cannot say that the limit is equal to its supremum, since it doesn't have a limit. However, this is an odd and less powerful phrasing of this theorem, which usually states that if the sequence is increasing and bounded above, then it converges to its supremum (and similarly for decreasing and bounded below). That is, bounded $\implies$ convergence, which your version doesn't give.
The question is whether rising without bound is "converging to $\infty$" or "diverging to $\infty$". This depends on the context you are working in. When working strictly in the real numbers $\Bbb R$, sequences that rise without bound do not have a limit, because $\infty \notin \Bbb R$. Thus the phrasing of the monotone convergence theorem.
However, you can work in the extended reals $\overline {\Bbb R} = \Bbb R\cup \{-\infty, \infty\}$. There the monotone convergence theorem still holds, but every sequence is bounded (by $-\infty$ and $\infty$). Thus in this context, the monotone convergence theorem simply states that all monotone sequences converge to their extremums.