Let $(x_n)$ be a real sequence, and $S_j := \sum_{i=1}^nx_i$ its partial sum, how can I formally justify that, if $(\sum_{i=1}^nx_i)$ converges,
$\lim_{m\to\infty}m^{-1}\sum_{i=1}^mS_i=\lim_{n\to\infty}\sum_{i=1}^nx_i$ ?
I can understand the intuition, but the next step - writing the proof - is not advancing properly.
As stated in a comment: what you are asking is essentially a standard result on the Cesàro mean (which is a specific case of the more involved Stolz—Cesàro theorem):
Theorem (Cesàro's Lemma). Let $(a_n)_{n\geq 1}$ be a real- or complex-valued sequence, and let $(s_n)_{n\geq 1}$ be the sequence of its Cesàro means: namely, $s_n = \frac{1}{n}\sum_{k=}^n a_k$ for $n\geq 1$.
Then, if $(a_n)_{n\geq 1}$ converges, then so does $(s_n)_{n\geq 1}$; and we then have $\lim_{n\to \infty} s_n = \lim_{n\to \infty} a_n$.
You want to apply this result to your sequence $(S_n)_{n\geq 1}$.