Epsilon-limit proof for real number sequences

[LCKurtz]

I think this would work.
Take N=max{N₁,N₂}
where N₁ is as you defined
and N₂ = the smallest integer greater than or equal to 2 |(a1-L)+...+(aN-L)| / ε.
If n > N, then |[(a₁+a₂+...+a_n)/n] - L|< ε.

That is correct.

However, according to what you said in post #7, we should take N to be the max of THREE things, i.e. N=max{N₁,N₂,N₃}. Why? What is N₃? I don't understand where this third restriction comes from.
I believe |[(a_N+1 - L) +...+ (a_n-1-L) +(a_n-L)] / n| <= (1/n)[|a_N+1 - L| + ...+ |a_n-1-L| + |a_n-L|] < (1/n)(n-N)ε/4 < ε/4 is always true, isn't it? So what is N₃?

In our back-and-forth we may have called N different things; that's why I summarized it in post #29. We don't need an N₃.