Is there any theorem like, that the limit of the average value of an infinite series takes the same value as the original sequence?

Question

Let lim n->infinity (an) = a be given(i.e. converges)

Then the sequence bn is defined as follows,
lim n->infinity (a1+a2+.......+an)/n

We need to comment on the convergence/divergence of bn.

So, lim n->infinity (a1+a2+.......+an)/n = lim n->infinity (an/n)= [lim n->infinity (an)]/[lim n->infinity (n)] = a/infinity ->0
==>bn converges

Or is there anything such as that the limit of the average value of an infinite series takes the same value as the original sequence?