Converge

In a metric space (X, d), a sequence $a_{n}$ in X is said to converge to a point $x$ if roughly speaking as $n$ goes to infinity $a_{n}$ gets closer and closer to $x$ and stays there. Rigorously, $a_{n}$ is said to converge to $x$ if for all $\epsilon >0$ there exists N such that for all n > N we have $d\left(a_{n},x\right)<\epsilon$.

Similar definitions can be made for convergence of functions.