So in real analysis we are given the definition for the convergence of a sequence:
The sequence $a_n$ converges to its limit $L$ if $\forall \epsilon >0$
$\exists$ an $N$(dependent on $\epsilon$) such that $n>N$ which
implies that $|a_n-L|<\epsilon$.
I know that $\epsilon$ is like the error factor but I don't understand what $N$ is. All I know is that it $\in$ $\Bbb{N}$.
Can someone elaborate?
Best Answer
$N$ is a number beyond which you know that the sequence differs from its limit by at most $\varepsilon$. In the figure below, we are given a sequence and an $\varepsilon$ that should form horizontal lines that bound the sequence for every point to the right of $N$.
If we can show that for all $\varepsilon>0$, we can find an $N$ that allows the sequence to the right of $N$ to be bounded, then the sequence is convergent.
Bear in mind, you have an error in your definition of convergence, as oernik has pointed out.
Image credit: http://math.feld.cvut.cz/mt/txta/2/txe3aa2a.htm