Prove an inequality with the help of bernoulli equality and archimedian property. I would like some help with the question below :
We have:
0 < q < 1.
Prove that for all ε > 0 there is M ∈ N (Natural number) , so for all n > M :
q^n < ε.
Hint:
Use Bernoulli Inequality and Archimedian Property.
For all those not sure what the hint is about :
Bernoulli Inequality :
Bernoulli
Archimedian Property :
For every ε > 0 there is a n ∈ N : (1/n) < ε
I'm thankful for any kind of help. Have a nice day!

Let $\frac{1}{q}=1+t$. Then $t$ is positive. By the Bernoulli inequality, if $n\gt 0$ we have
$$q^n=\frac{1}{(1+t)^n}\le \frac{1}{1+nt}\lt \frac{1}{nt}.$$
We want to make this $\lt \epsilon$.  Equivalently, we want $n\gt \frac{1}{t\epsilon}$. The Archimedean property tells us that there  is an integer $M$ such that $M\gt \frac{1}{t\epsilon}$.  And if $n\ge M$, it is clear that $n\gt \frac{1}{t\epsilon}$.
