Let f: D-->R and let c be an accumulation point of D. Suppose that a<=f(x)<=b for all x elements in D with x not equal to c, and that lim(x-->c)f(x)=L. Prove that a<=L<=b.
Suppose L<a ===> a-L>0............................................... ...........1
Now since , we HAVE that for every +ve NO (and thus for a-L) ,THERE exists a δ>0 and such that:
if xεD and 0<|x-c|<δ,then |f(x)-L|<a-L.
But : |f(x)-L|<a-L <====> L-a<f(x)-L<a-L ====> f(x)<a ,a contradiction.
In the same way we work for b