Let $\{a_n\}$ a sequence converging to $L > 0$. Show that $\exists N \in \mathbb{N}$,$\forall n \in \mathbb{N}$, $n \geq N$, $a_n >0$.

Question

I know that for any $\epsilon>0$, $\exists N \in \mathbb{N}$, $\forall n \in \mathbb{N}$, $|a_n -L|<\epsilon$ and the hint we were given is $-\epsilon <a_n-L< \epsilon$.

I don't understand where to start.

Answer 1

To solve this problem, you should be familiar with several ways of expressing $|a_n−L|<ϵ.$

$|a_n−L|<ϵ \iff −ϵ<a_n−L<ϵ \iff L−ϵ<a_n<L+ϵ$

Now, let $\epsilon = L/2$. You have that $0<L/2=L−ϵ<a_n<L+ϵ$

Answer 2

Hint

Take $\varepsilon =\frac{L}{2}$ in the definition.