Prove the equivalence between the existence of a limit and a corollary about the limit's vicinity.

Question

Prove these are equivalent:

a. $\lim_{n\to \infty}{a_n}=L$

b. Every vicinity of $L$ contains all the sequence's elements except for a finite number of elements.

I started proving $a. \Rightarrow b.$ but now I see that I assumed that every vicinity of $L$ is $U_{\delta}(L)=(L-\delta,L+\delta)$ and that way I managed to prove it. Assuming it is not necessary, how can I then prove it?

Answer 1

a. implies that $\forall n>n_0 \ |a_{n >n_0}-L|<\varepsilon $. Since $n$ are increasing and unbounded, only terms $a_0, \ldots a_{n_0}$ are outside of the $\varepsilon-$neighborhood/radius of $L$ with $n_0 = n_0 (\varepsilon)$.