accumulation points,limit points,cluster points are one and the same thing .Any book in analysis will tell you that
accumulation points,limit points,cluster points are one and the same thing .Any book in analysis will tell you that
That is demonstrably not true. Unfortunately, authors differ in their definitions of limit point (or accumulation point). Some authors distinguish between a limit of a sequence and a limit point of a set. That must be true in particular for the author of the problem at the start of this thread.
Some (but by no means all) authors define a limit point of a set in such a way as to exclude an isolated point from being a limit point. But when defining a limit point of a sequence, it is essential to allow an isolated point to count as a limit, otherwise a constant sequence would not converge. Lang's definition in comment #3 above refers to limit points of sequences, so of course it allows isolated limit points. I have no idea whether Lang uses the inclusive or the exclusive definition for limit points of sets in metric spaces.
The Wikipedia definition for a limit point of a set is uncompromisingly exclusive: "In mathematics, a limit point (or accumulation point) of a set S in a topological space X is a point x in X that can be "approximated" by points of S other than x itself."
There is agreat difference between the definition
1)of the limit of a sequence and the definition
2) of the limit point of a sequence.
1 IMPLIES 2 and this is provable as i showed ,while 2 does not imply 1
check the two definition again