I think you are looking for \ge and \le. \gt and \lt are not defined.
\documentclass[varwidth]{standalone}
\usepackage{amsmath}
\begin{document}
\begin{equation*}
\forall n,m \ge n_{\varepsilon} \quad \Longrightarrow \quad ||\mathbf{x}_n - \mathbf{x}_m|| \le \varepsilon
\end{equation*}
\end{document}

Edit 1
The > and < signs are already on your keyboard:
\documentclass[varwidth]{standalone}
\usepackage{amsmath}
\begin{document}
\begin{equation*}
\forall n,m > n_{\varepsilon} \quad \Longrightarrow \quad ||\mathbf{x}_n - \mathbf{x}_m|| < \varepsilon
\end{equation*}
\end{document}

In case you want to use \gt or \lt, you have to define it. The definition below is taken from this answer:
\documentclass[varwidth]{standalone}
\usepackage{amsmath}
\newcommand{\lt}{\symbol{"3C}}
\newcommand{\gt}{\symbol{"3E}}
\begin{document}
\begin{equation*}
\forall n,m \lt n_{\varepsilon} \quad \Longrightarrow \quad ||\mathbf{x}_n - \mathbf{x}_m|| \gt \varepsilon
\end{equation*}
\end{document}

>and<? – Mar 25 '19 at 10:35\gtand\ltare not already defined. I think you are following this answer or something similar, but you should make sure that\newcommands are included. IMHO you should just use>and<. We should not make things more complicated. – Mar 25 '19 at 10:43