I don\'t know where to begin with this one. Is this how I begin? (Help write up

Question

I don't know where to begin with this one.

Is this how I begin? (Help write up this proof).

A sequence {an} in a metric space (X, d) is said to converge if there exists a %u2208 X such that for every %u03B5 > 0 there exists a real number N%u03B5 such that for all n %u2208 N and n %u2265 N%u03B5, d(an, a) < %u03B5.

Explanation / Answer

Since this is an iff proof, you want to break it into two parts: a) If lim n->infinity a_n=p, then lim n->infinity d(a_n,p)=0. I think this direction is trivial; observe that d(a_n,p)->d(p,p)=0 as n->infinity. b) If lim n->infinity d(a_n,p)=0, then lim n->infinity a_n=p. Perhaps proof by contradiction works. Suppose by contradiction that lim n->infinity a_n=q, some number not equal to p. Then lim n->infinity d(a_n,p)=d(q, p) is non-zero. But we know this quantity to be 0 by assumption. Thus, we have a contradiction.

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