# Sets & limit points and stuff

1. Sep 27, 2005

### *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?

2. Sep 27, 2005

### fourier jr

you need the definition of convergence also

3. Sep 27, 2005

### 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)