This is an exercise from Gromov's Partial differential relations. (page 5)

Let $V$ and $V'$ be two complex submanifolds in $\mathbb{C}^N$ of complimentory dimension. Prove that $V$ and $V'$ intersect if the following sets are compact

$V_k = \{(v,v') \subset V\times V'|dist(v,v') \leq k$}.

I was looking for a differential geometry approach to solving this problem along the lines of Theorem 2 of Frankel(1961) but anything would do.