Let Y be non empty subset of R which is bounded above and y $=$ supY then show that y $ \in Y \cup$ Y', where Y' is set of limit points of Y
Now sup exists by lub property. If sup belongs to set Y then i am done. if not then there exists an element a $\in$ Y such that $a > y- \epsilon$. so a $\in$ $(y-\epsilon, y + \epsilon$ ) for any spsilon. So neighbourhood of y has a element a. So it is limit point.
Is this correct?