Proving the Existence of Limit Points through Convergence of Sequences

[*melinda*]

The question says:

Let A be a set and x a number.
Show that x is a limit point of A if and only if there exists a sequence x_1 , x_2 , ... of distinct points in A that converge to x.

Now I know from the if and only if statement that I need to prove this thing both ways.

So, the proof in one direction (I think) would be that I have a limit point x\in A, and would need to construct a sequence that converges to x.
Why are these things always easier said than done ?

One of the definitions in my book states:
x is a limit point of A if given any error 1/n there exists a point y_n of A not equal to x satisfying |y_n -x|<1/n or, equivalently, if every neighborhood of x contains a point of A not equal to x.

I feel like I can somehow use this definition, or at least the definition of Cauchy sequences to help with my proof. Only trouble is, I don't know what to do with what I have.

help?

 

[fourier jr]

you need the definition of convergence also

 

[Tzar]

Cauchy is irelevant here. Use your defenition of limit point to select an element y_n in A for each n. as n goes to infinity, y_n goes to x since 1/n goes to zero. And you're done. (one way)