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

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

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

I was looking for a differential geometry approach to solving this problem along the lines of Theorem 2 of [Frankel](http://projecteuclid.org/download/pdf_1/euclid.pjm/1103037541)(1961) but anything would do.