Statement to prove:

If A is a countable subset of an uncountable set X, prove

that X \ A (or "X remove A" or "X - A") is uncountable.

My work so far:

Let A be a countable subset of an uncountable set X.

(N denotes the set of all naturals)

So A is equivalent to N (or "A ~ N") by the definition of countable.

X is not equivalent to N since X is uncountable.

Assume X \ A is countable. That is (X \ A) ~ N. Thus there is a bijection from (X \ A) to N. However since X is uncountable it is not guaranteed there is an n in N such that there is an x in X that maps to n. And so X \ A is uncountable.

I know that proof is incomplete and may even be completely off :|

I am really struggling in this class (Intro to Real Analysis) and feel as if it will be my GPA-killer (but I don't want it to be!). I'm having a hard time with the countable/uncountable concepts we covered last week.

Any help is greatly appreciated! Thank you for your time.