Thread: Limit Point Proof.

Suppose that A and B are non-empty sets of real numbers, and that x is a limit point of A union B. Prove that x is a limit point of A or of B.


=> I think that I need to do proof by cases for this one, and use the epsilon neighborhood in some way, but I am not sure.

Also, could this be more easily proved using proof by contradiction,
assuming that x is not a limit point of a and x is not a limit point of B.

Thanks for you help.

Originally Posted by eg37se

Suppose that A and B are non-empty sets of real numbers, and that x is a limit point of A union B. Prove that x is a limit point of A or of B.


=> I think that I need to do proof by cases for this one, and use the epsilon neighborhood in some way, but I am not sure.

Also, could this be more easily proved using proof by contradiction,
assuming that x is not a limit point of a and x is not a limit point of B.

Thanks for you help.

So there exists a sequence {x_n} in A \/ B s.t. x_n --> x . For any n, x_n belongs either to A or to B. It can't be that both A and B contain a finite number of elements of the sequence ==> either A or B (or both, of course, as "or" in mathematics is always inclusive unless otherwise stated) contain an infinite number of elements of {x_n} ==> since {x_n} converges so does any infinite subsequence and to the same limit ==> we're done.

Tonio

I considered this way too, I just wasn't sure. Thank you for the help.