Let $X,Y$ be complex manifold, $Y$ Stein. It sounds quite reasonable to formulate the following claim: given $y_0\in Y$, for every $\epsilon>0$ and $M\subset X$ compact, there exists an holomorphic mapping $h\colon X\to Y$ with dense image (that is $\overline{h(X)}=Y$) such that $$ \|h-y_0\|_M<\epsilon $$ (the distance is taken with respect some Riemannian metric on $Y$).

This sounds like a very standard result; can somebody provide a reference? Or suggest some counter example, but I think this fact holds true. Thanks