Continuous Mappings of Dense subsets

Hello,

This is a question from Baby Rudin, Exercise 4.4 I was wondering if anyone could verify my reasoning:

The question is, if is a continuous function from a metric space to a metrix space , and a set is dense in X, then show in dense in .

So I have the solution:

Since is dense in , then for any , we can construct a sequence of points where the and .

Since f is continuous, we know , and since and , is dense in .

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So my qualms with that solution are

a) The first statment,

Since is dense in , then for any , we can construct a sequence of points where the and .

Is that a true statement for all dense subsets of a metric space ?? it seems like a serious claim to make, I have never heard or seen a theorem that says anything like that and I can't find anything in Rudin.

b) The last statement,

Since and , is dense in .

How does converging to show us that is dense in ? Is it because is arbitrary? is the theorem (which I don't know about) an if and only if theorem?

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Essentially I am asking if there is a theorem (which I can't seem to find anywhere) which states:

is dense in , if and only if for any , we can construct a sequence of points where the and .

Thanks, any help appreciated!!