We define a two player game in $\mathbb{R}$. Player 1 chooses an uncountable set $X_1 \subseteq \mathbb{R}$. Player 2 chooses some uncountable subset $X_2 \subseteq X_1$. The game continues and a chain of sets $X_1 \supseteq X_2 \supseteq X_3 \supseteq \dots$ is formed. Player 1 wins if $\bigcap X_n$ is nonempty and Player 2 wins otherwise. Prove that Player 2 has a winning strategy.
Hints? I managed to make intersection consist of at most one point (for example there should be some $[n,n+1]$ with uncountably many points of $X_1$, so you can get $X_2 \subseteq [n,n+1]$. Now one of $[n, n+1/2]$ or $[n+1/2,n+1]$ should have uncountably many points of $X_3$ and so on...) but I can't make the argument better.