Converge

In a metric space (X, d), a sequence $a_{n}$ in X is said to converge to a point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon>0} there exists N such that for all n > N we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\left (a_n, x\right ) < \epsilon } .

Similar definitions can be made for convergence of functions.