Limit of a sequence
The intuitive concept of the limit of a sequence is very simple. The terms in the sequence $\frac{1}{n}$ get closer to $0$ as $n$ gets larger so the limit is $0$. However the phrase ‘get closer’ is not very mathematical and in fact is misleading. (The sequence $0,0,0,0,\dots $ has limit $0$ but in what sense are the terms of the sequence getting closer to $0$?)

The definition of limit took mathematicians many years to develop — in fact, it took many years for them to realise that a good definition was needed — and so it is not surprising that it is a hard concept to understand.

Definition
The number $a$ is a limit of the sequence $(a_n)$ if for all $\varepsilon >0$, there exists $N\in \mathbb{N}$ such that for all $n> N$ we have $|a_n-a|<\varepsilon $. We write $a_n\to a$ or $\displaystyle \lim_{n\to \infty } a_n$ and say '$a_n$ tends to $a$, or $(a_n)$ converges to $a$, as $n$ tends to infinity’.

If a sequence $(a_n)$ has a limit, then we say that $(a_n)$ is convergent.

Remark
Some authors use $n\geq N$ rather than $n>N$. It makes no difference to the theory.

The above definition of limit is certainly far from intuitive but understanding it is vital for the understanding of sequences and series. Let’s do some examples.

Example
The limit of $a_n = 1/n$ is $0$.
Proof
Let $N$ be a natural number greater than $1/\varepsilon $. Then, $1/N< \varepsilon $ and we have \begin{align*} |a_n - a | &= | 1/n - 0 | \\ &= |1/n| \\ &= 1/n \\ &\leq 1/N , {\text{ for }} n\geq N, \\ &< \varepsilon . \end{align*} In this example, note that it is not obvious why we should choose $N$ to be greater than $1/\varepsilon$. This is a common point of confusion and we will come back to it later.

Remark
The type of proof used above to show that $1/n\to 0$ is called an $\varepsilon-N$ proof.