# Prove that the following sequence is convergent

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?

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.