Every convergent sequence is Cauchy

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I just did this to get the ball rolling. Page is of low grade due to ease of proof.

Statement

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

• $\Big(\forall\epsilon>0\ \exists N\in\mathbb{N}\ \forall n\in\mathbb{N}[n>N\implies d(a_{n},a)]\Big)\implies$$\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)$

Proof

(Unknown grade)

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The message provided is:

Easy proof, did it in my first year

See also

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TODO: This too

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