Difference between boundary point & limit point. A limit point is just a accumulation point whose neighbourhood contains infinitely many elements of the sequence. 
Is there any difference between boundary point & limit point? I've read in another question here that all boudary points are limit points, but is the converse true?
 A: Consider the interval $[0,1]$.  Each element of it is a limit point, i.e. $\alpha$ is a limit of the sequence $n_1=\alpha, n_2=\alpha, \ldots$.  Only $0,1$ are boundary points.
A: Definition of Limit Point: "Let $S$ be a subset of a topological space $X$. A point $x$ in $X$ is a limit point of $S$ if every neighbourhood of $x$ contains at least one point of $S$ different from $x$ itself."~from Wikipedia
Definition of Boundary: "Let $S$ be a subset of a topological space $X$.  The boundary of $S$ is the set of points $p$ of $X$ such that every neighborhood of $p$ contains at least one point of $S$ and at least one point not of $S$."
~from Wikipedia
So deleted neighborhoods of limit points must contain at least one point in $S$.  But (not necessarily deleted) neighborhoods of boundary points must contain at least one point in $S$ AND one point not in $S$.
So they are not the same.
Consider the set $S=\{0\}$ in $\Bbb R$ with the usual topology.  $0$ is a boundary point but NOT a limit point of $S$.
Consider the set $S'=[0,1]$ in $\Bbb R$ with the usual topology.  $0.5$ is a limit point but NOT a boundary point of $S'$.
A: Well, as someone has figured it out by supplying the definitions of limit point and boundary point. Now if we just head toward the general set topological approach we will find that , if  $\Bbb{S}$ ${\subset}$ of  $\Bbb{R}$ , and  if  $\Bbb{X}$ be the boundary then 
 $\Bbb{X}$=cl(S)~int ( S) .
So if p is a boundary point, then p will be in  $\Bbb{X}$  . And we call        $\Bbb{S}$  a closed set if it contains all it's boundary points.
Now as we also know it's equivalent definition that s will be a closed set if it contains all it limit point.
But that doesn't not imply that a limit point is a boundary point as a limit point can also be a interior point . Let's check the proof.
Let  $\Bbb{S}$  is our set of which l is a int  point . Then for `$\epsilon$>0 , N(l, $\epsilon$ ) contained in l . Now we will try to prove it contrapositively .
 Let  l is not an int point . Then N(l, $\epsilon$ ) is not contained in  $\Bbb{S}$ . Now let, €>0  then either €< $\epsilon$ or €≥ $\epsilon$
When, €<  $\epsilon$  as N(l, $\epsilon$ ) is not contained in  $\Bbb{S}$ , so N(l, €) is not also contained in s . It suggests that,
N'(l,€)  ${\cap}$ $\Bbb{S}$  = $\phi$
So, l is not a limit point of  $\Bbb{S}$
When, €≥  $\epsilon$ , N(l, $\epsilon$ ) is contained in N(l, €). So from here also it can be shown that ,  $\Bbb{S}$ ${\cap}$  N'(l,€) is  $\phi$ . So l is not a limit point of  $\Bbb{S}$ .
So if l is not an  int point of  $\Bbb{S}$  , it's not an limit point of  $\Bbb{S}$  . It implies that if l is an limit point of  $\Bbb{S}$  ,  it's an interior point of   $\Bbb{S}$   .
Now, there are also some cases where the above assertion fails. So l may or may not belongs to  cl ( $\Bbb{S}$  ) ~  int (  $\Bbb{S}$  ) 
And the whole discussion tells us that a limit point can be a boundary point but that doesn't mean every limit point is a boundary point. 
And that's it !!!
