Prove that the following sequence is convergent

Question

Prove that the following sequence is convergent? Prove using the theorem that if a sequence is bounded (below/above) and monotonic increasing/decreasing it is convergent.

$a_n=\frac{4n-9}{5n+8}$

(There are many similar questions but not exactly one like that)

1. My first difficulty is the next step in showing that is bounded. What I mean here is not a simple answer, just look at numbers, but mathematical proof(I plug the numbers in the sequence and I realised that inf=$-\frac{9}{8}$) $ | a_n | \leq M, \forall n$ how to take another step?

2. I verified that $ \lim_{a \rightarrow \infty} =\frac{4}{5}$

3. I tried to check that this sequence is increasing by $\frac{a_n}{a_(n+1)}\leq 1$ and I end up with that $20n^2+7n-40 \geq 0$ Is that proving that this sequence is increasing?

Thank you. Please be respectful

Answer 1

Your sequence is $$a_n=\frac45-\frac{77}{25x+40}$$ which is clearly strictly increasing. It is also bounded above by $\frac45$ and hence converges by the monotone convergence theorem.