Converge in expectation imply convergence in partial sum of expectation for iid Random variables

Question

I am stuck on the following step in a book.

Given that $X_1,X_2,...$ are iid random variable, and two sequences {Xn}, {Yn} are equivalent. Here is what the book says:

Clearly $\mathscr{E}\left(Y_{n}\right) \rightarrow \mathscr{E}\left(X_{1}\right)$ as $n \rightarrow \infty$; hence also $$ \frac{1}{n} \sum_{j=1}^{n} \mathscr{E}\left(Y_{j}\right) \rightarrow \mathscr{E}\left(X_{1}\right), $$

I am able to show $\mathscr{E}\left(Y_{n}\right) \rightarrow \mathscr{E}(X_n)$ but I don't see how this implies the second part. Any help is appreciated.

Answer 1

Here is my answer to my own question.

Intuitively, this implication work because for large n, $E(Y_n)$ is very close to $E(X_1)$, and the differences from the first finite many terms, after averaging over n, go to $0$.