Originally Posted by
southprkfan1 Let (xn) be a convergent sequence in E', (xn) --> x. We want to show x is in E'. I.e, we want to show x is a limit point of E.
We know x is a limit point of E if given any e>0 there is a point in E that is in the ball of radius e around x. That is, there exists a p in E such that:
lx - pl < e
Heres the proof:
Since (xn) --> x, then there is a k where l x-xk l < e/2 *
and since xk in in E', it is a limit point of E, so there is a point p in E such that: l xk - p l < e/2 **
I claim: lx - pl < e
Pf.
lx - pl <= lx-xkl +lxk - pl < e/2 + e/2 (by * and **)
so lx- pl < e