Assuming I understand the question, this is what I would do.

If (an) converges toathen that means that there exists an N in the Naturals such that if n is greater than N, |(an) -a| < d for some d > 0

Now there are a finite number of values more than d away froma, so the maximal element is either the maximum of these few values, or a+d. Similar for the minumum element.