Answer to Question #2108 in Real Analysis for Vijay

Question

Prove that if the sequence (Un) is convergent and bounded above by M, then the limit is bounded above by M

Answer 1

If the sequence is convergent and the limit exists, there exist such N,& that the value of& U(n) ( n > N) tends to limit. Since the sequence is bounded by M, each term |Un| of this sequence is less than M, so the limit cannot be more than M.