# Thread: Definition of a limit of a sequence

1. ## Definition of a limit of a sequence

Hi all,

Can anyone help me to understand the definition of a limit of a sequence. In notes and when reading about it on the internet, I always find for every ε > 0 ∃ n such that for every n ≥ N, mod(An - L) < ε.

My biggest problem is picturing this. I understand that epsilon is a value very close to the limit of the sequence but I don't understand how this all relates together to allow us to show that a sequence converges to its limit e.g. 1/n^1/2 -> 0.

Anybody able to explain this?

Thanks guys

2. ## Re: Definition of a limit of a sequence

Let's take your example of the sequence $\displaystyle a_n = \frac{1}{\sqrt{n}}$

The goal is to find an equation for N dependent on epsilon such that for $\displaystyle n \geq N$ $\displaystyle |a_n-L| < \epsilon$ If $\displaystyle L=0$, then

$\displaystyle |a_n-L| = |\frac{1}{\sqrt{n}}| = \frac{1}{\sqrt{n}} < \epsilon \iff n > \frac{1}{{\epsilon}^2} = N}$

so no matter how small epsilon (the interval) gets, we can always find a natural number $\displaystyle n \geq N$ such that a point $\displaystyle a_n$ is in that interval. As epsilon goes to 0, the difference between the points $\displaystyle a_n$ and $\displaystyle L$ becomes sufficiently small