If ((a(n))+(a(n+1)))/2 converges to A, then a(n) converges to A. Either prove or disprove this conjecture.
Normal convergence proof
The Attempt at a Solution
I will prove that ((a(n))+(a(n+1)))/2 converges to A such that for every (epsilon)>0 there exists a positive integer N such that for every n>N abs(a(n)-A)<(epsilon).
Consider (epsilon)> 0 arbitrary.
Because a(n) and a(n+1) will converge to the same number...
That second part is what I am stuck at. I am really good at proving things converge this way with sequences defined but struggle in the abstracts. I struggle with the second and fourth lines of the proof, as the third is just consider n>N arbitrary.