Q: Suppose that A and B are non-empty sets of real numbers and that x is a limit point of AUB. Prove that X is a limit point of A or of B.

my solution:

Given x is a limit point of AUB

then for all epsilon > 0, nbd(x) contains a point of AUB different from x

Then nbd(x) contains a point of A or nbd(x) contains a point of B different from x.

Therefore x is a limit point of A or of B.

[nbd = neighborhood]

Just wondering if my approach is correct or not. I feel it's not as easy as I thought...

help please thanks!