Let \(X\) be a topological space and \((x_n)\) a sequence on \(X\). The sequence converges to some \(x\in X\) if for every neighbourhood \(U\subset X\) there is a \(n_0\in \mathbb{N}\) such that \(x_n\in U\) for all \(n\ge n_0\).
The point \(x\) is called limit point or simply limit of \((x_n)\).
We write \(\lim_{n \to \infty}x_n=x\), \(x_n\xrightarrow{n\to \infty} x\) or simply \(x_n\to x\).
We call non-convergent sequences divergent.
Special sequences Link to heading
- \(\sqrt[n]{n}\to 1\) (proof )