Is this an equivalent way of writing supremum

Question

Let $\{A_n\}$ be a sequence of events I have an indicator r.v. $I_n$ where $I_n=1$ if an event $A_n$ occurs and $0$ otherwise

Let $X_n = \sum_{k=1}^n I_k$

Is it the same thing to say this

$\sup_{n\in \{1,2,3...\}} E(X_n)$

is equal to

$\lim_{n\rightarrow \infty} E(X_n)$

Answer 1

Yes. $\{X_n\}$ is an increasing sequence so $EX_n$ is also increasing. Hence $lim_n EX_n=sup_n EX_n$ by monotonicity.