Limit point need not belong to a set

Question

I was reading multivariable calculus , where there were a few topological definitions on things like open ball, limit points, boundaries etc.i came across the statement "a limit point of a set need not belong to the set". Why is this true? can someone give a visual and mathematical proof of this.

Answer 1

When we talk about limit point of a set we mean some point in the Metric Space around whose neighbourhood there are infinitely many points of the set located.

To be more precise, no matter how close the boundary of that point is considered there will be some point(rather infinitely many) of the set within the boundary.

Mathematically,

if $A \subset X$ then $y \in X$ is said to be a limit point of $A$ if for each $\epsilon$ >0

$N(y,\epsilon)\cap A\setminus\{y\} \neq \phi$.

For example

Consider $\{\frac{1}{n} : n \in N\}$

This has limit point $0$ which does not belong to the set.