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. |
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[[category: Topology]] | [[category: Topology]] | ||
Revision as of 20:13, September 12, 2008
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 functions.