If you need more details I can write down all the $\epsilon, \delta$ 's

Its a bit messy to write down, but the idea is that if the interval $ [m_k, m_{k+1}) $ is much longer than the interval $[1,m_k)$, when you average the only part of the sequence that counts is in the r.v. having index in the longer interval.

@FedorPetrov You are right, even easier !

I was probably skipping details, but since the series convergences pointwise the limit must coincide with the norm limit (in any reasonable norm like L2)

@ZhangYuhan This case can be reduced to the previous one, see the edit