Use the definition of convergence of a sequence to show the sequence $a_n=4(\frac{3}7)^n +9$ converges.

Question

I know the definition of convergence, I know how to start off proving a sequence converges. On this sequence,

$a_n = 4 (\dfrac{3}7)^n +9$, $n \in \mathbb N$.

I don't understand how to show epsilon is less then the mod of the sequence takeaway the limit of the sequence.

Answer 1

With Bernoulli's inequality $$a_n-9=4\left(\frac37\right)^n=\dfrac{4}{(\frac73)^n}=\dfrac{4}{(1+\frac43)^n}\leq\dfrac{4}{1+\frac{4n}{3}}<\dfrac{3}{n}$$