Lemma 2

Let X be a topological space. Let F and D be disjoint subspaces of X such that F is closed and D discrete. If F contains no nonempty perfect subspace, the same is true of $F\cup D$.

Suppose indeed that K is a nonempty perfect subspace of $F\cup D$. Let $x$ be a point of K. If $x$ belongs to D, it is isolated in D, hence also in $F\cup D$ since F is closed. Therefore $x$ is isolated in K, which contradicts the assumptions.