Difference between revisions of "Every convergent sequence is Cauchy"

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Revision as of 16:16, 13 March 2016

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I just did this to get the ball rolling

Statement

If a sequence [ilmath](a_n)_{n=1}^\infty[/ilmath] in a metric space [ilmath](X,d)[/ilmath] converges (to [ilmath]a[/ilmath]) then it is also a Cauchy sequence. Symbolically that is:

  • [ilmath]\Big(\forall\epsilon>0\ \exists N\in\mathbb{N}\ \forall n\in\mathbb{N}[n>N\implies d(a_{n},a)]\Big)\implies[/ilmath][ilmath]\Big(\forall\epsilon>0\ \exists N\in\mathbb{N}\ \forall n,m\in\mathbb{N}[n\ge m>N\implies d(x_n,x_m)<\epsilon]\Big)[/ilmath]

Proof

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Easy proof, did it in my first year

See also

TODO: This too

References

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