Every convergent sequence is bounded.
Proof: To prove the result we need to find the bound. First, assume \(x_{n}\to x\) as \(n \to \infty\) hence \(\forall \epsilon >0,~\exists N\in \mathbb{N}:\forall n\geq N,~|x_{n}-x|<\epsilon\). From the triangle inequality we have that \[ |x_{n}|= |x_{n}-x+x|\leq|x_{n}-x|+|x|\leq \epsilon +|x|. \] Hence the bound \(M\) can be defined as \[ M=\max(|x_{1}|, \dots, |x_{{N-1}}|, |x|+\epsilon ). \]