Cantor's intersection theorem and Baire Category Theorem From an old post in math stackexchange, I read a comment which goes as follows " I like to think of Baire Category Theorem as spiced up version of Cantor's Intersection Theorem". My question -----is it possible to derive the latter one using the former?

Do you have a copy of Rudin's Principles of Mathematical Analysis? If you do, then problems 3.21 and 3.22 outline how this is done. Quoting here:

3.21: Prove: If $(E_n)$ is a sequence of closed and bounded sets in a complete metric space $X$, if $E_n \supset E_{n+1}$, and if $\lim_{n\to\infty}\text{diam}~E_n=0$, then $\cap_1^\infty E_n$ consists of exactly one point.
3.22: Suppose $X$ is a complete metric space, and $(G_n)$ is a sequence of dense open subsets of $X$. Prove Baire's theorem, namely, that $\cap_1^\infty$ is not empty. (In fact, it is dense in $X$. Hint: Find a shrinking sequence of neighborhoods $E_n$ such that $\overline{E_n}\subset G_n$, and apply Exercise 21.

Proving density isn't actually much harder than proving nonemptiness.
