My Approch: If $X$ is countable complete metric space, it has atleast one isolated point $x$ by Baire Category Theorem. Consider, $X_1$= $X\setminus \{x\}$ Since,$x$ is isolated point of $X$, hence $X_1$ is also complete & countable. By similar argument we can consider $X_2$ & so on.
Is this correct approach to conclude $X$ has infinitely many isolated points.