Let $X_t$ be a submartingale and $\tau_n$ a sequence of stopping time. Therefore $E[X_{\tau_n}]$ is an increasing sequence. In a proof that I'm reading I see the following:
Since $E[\sup_t|X_t|]<\infty$ (is an assumption), then $(E[X_{\tau_n}])_{n\in \mathbb N}$ is $\underline {also}$ bounded and $(E[X_{\tau_n}])_{n\in \mathbb N}$ is convergent.
What I don't understand is why $E[\sup_t|X_t|]<\infty$ imply that $(E[X_{\tau_n}])_{n\in \mathbb N}$ is bounded? I always thought that being $<\infty$ was different from being bounded, but this sentence with the use of "also" make me uncertain about it.
$|X_{\tau_n}| \leq \sup_t |X_t|$, which proves $E[X_{\tau_n}] \leq E[\sup_t |X_t|]<\infty$ for all $n$.
You're right that just having $E[X_{\tau_n}]<\infty$ for each $n$ does not imply the entire sequence is bounded. However, notice that we have the same upper bound for each $n$, namely $E[\sup_t|X_t|]$. This is enough to prove the sequence is bounded.