Proving Convergence of Real Number Sequences with Metric Equations

[muzak]

Homework Statement

Prove that lim_{n} p_{n}= p iff the sequence of real numbers {d{p,p_{n}}} satisfies lim_{n}d(p,p_{n})=0

Homework Equations

The Attempt at a Solution

I think I can get the first implication. If lim_{n} p_{n}= p, then we know that d(p,p_{n}) = d(p_{n},p) <\epsilon. Then given \epsilon > 0 and some N, for n>N we have |d{p,p_{n}-0|<d{p,p_{n} = d(p_{n},p) &lt; \epsilon.

I'm having a little trouble with the backwards implication, do I just do what I did up above but backwards sorta? Or should I pick some p_{n} and show that it converges to 0, like 1/n or something.

 

[CompuChip]

You are very much on the right track, but your "proofs" are still quite sloppy and it's hard to see whether you're reasoning circularly here.
For example,

I think I can get the first implication. If lim_{n} p_{n}= p, then we know that d(p,p_{n}) = d(p_{n},p) <\epsilon.

Actually, you know that for any \epsilon &gt; 0 there is an N such that this is true for all n > N.

Maybe it helps if you first write out exactly what you need to prove, in the form:

For all \epsilon &gt; 0, I need to prove that ...(if there exists an N such that ... there also exists an N' such that ...)... , and that (the other implication).