A real sequence \(\{ x_{n} \}_{n\in\mathbb{N}}\) where for all \(n \in N\), \(x_{n} \in \mathbb{R}\) converges to some limit \(x \in \mathbb{R}\) as \(n\to \infty\) (denoted \(x_{n } \to x\)) when \[\forall \epsilon >0, \exists N\in \mathbb{N} :\forall n\geq N, |x_{n}-x|<\epsilon.\]
Proof Template - Let \(\epsilon>0\) be arbitrary. - Try to solve the inequality \(|x_{n}-x|<\epsilon\) to determine how to choose \(N \in \mathbb{N}\) so that this inequality holds for all \(n \geq \mathbb{N}\). - Show that the \(N\) found actually works, namely for all \(n \geq N\).
Example To demonstrate how we use this result to show convergence we show that \(x_{n}=\frac{1}{n}\) converges to \(0\) as \(n \to \infty\). Fix \(\epsilon>0\) and find \(N\) such that for all \(n\geq N\) \[ \left| \frac{1}{n}\right|=\frac{1}{n}<\epsilon \implies N> \frac{1}{\epsilon }. \]