Difference between revisions of "Converge"
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| − | In a [[metric space]] (X, d), a sequence <math> a_n</math> in X is said to '''converge''' to a point <math>x</math> if roughly speaking as <math> n </math> goes to infinity <math> a_n</math> gets closer and closer to <math>x</math> and stays there. Rigorously, <math> a_n</math> is said to converge to <math>x</math> if for all <math>\epsilon>0</math> there exists N such that for all n > N we have <math>d\left (a_n, x\right ) < \epsilon </math>. | + | In a [[metric space]] (X, d), a [[sequence]] <math> a_n</math> in X is said to '''converge''' to a point <math>x</math> if roughly speaking as <math> n </math> goes to infinity <math> a_n</math> gets closer and closer to <math>x</math> and stays there. Rigorously, <math> a_n</math> is said to converge to <math>x</math> if for all <math>\epsilon>0</math> there exists N such that for all n > N we have <math>d\left (a_n, x\right ) < \epsilon </math>. |
| − | Similar definitions can be made for convergence of | + | Similar definitions can be made for convergence of [[function]]s. |
| − | [[Category | + | [[Category:Topology]] |
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Latest revision as of 18:37, June 30, 2016
In a metric space (X, d), a sequence 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 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 n } goes to infinity 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 a_n} gets closer and closer to 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} and stays there. Rigorously, 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 a_n} is said to converge to 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 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.