Yeah, I was just asked to prove this using nothing but the definition of convergence. As I struggled with it, I came across this thread and decided to post what I got.
Here's how I did it. (There are several ways, but I think this way is the most direct.)
x_n-->L & y_n=(x_1+...+x_n)/n ==> y_n-->L
PROOF:
x_n-->L, so L-e<x_n<L+e for all n>N.
So, x_n/(L-e)>1 and x_n/(L+e)<1.
Now, let y_n
=[x_1+...+x_(N-1)]/n + [x_N+...+x_n]/n.
The first term...