Question

Let {an}∞n=1 be a convergent sequence with limit L such that an ≤ B forevery n≥N for some N ∈ N and some B ∈ R. Prove that L ≤ B

Answer 1

We have that $\lim \limits_{n \to \infty} a_n = L$

Also, $a_n \leq B$

Taking limits we have $\lim \limits_{n \to \infty} a_n \leq \lim \limits_{n \to \infty} B$

Hence $L \leq \lim \limits_{n \to \infty} B = B$

As desired