Limit of a sequence (Calculus 1)

Why is the following statement True ?

limn→+∞ an = L then limn→+∞ 1/n Σk=1n ak = L

If you're using epsilon-delta definitions, here's a proof:

For e>0 there's an N such that for k>N, |ak - L|<e/2.

Take M = |(a1-L) + (a2-L) + .... + (aN-L)|, then for k>N

|a1+...+ak - k*L| <= M + (k-N)e/2

dividing by k

|(a1+....+ak)/k - L| <= M/k + ((k-N)/k)(e/2)

and in the limit k->inf that's e/2, so there's some N' where k>N' implies that's <e.

That means for the given e>0, that N' satifies the limit definition for

lim (a1+...+ak)/k = L