I am studying the proof of Remmert's Theorem on the book Griffiths & Harris - Principles of Algebraic Geometry, Chapter 3 Section 2 page 395. There is this observation that I do not understand:
In $\mathbb{C} \times \mathbb{C}^p \times \mathbb{C}^q$ with coordinates $(x, v_1, \ldots, v_p, w_1, \ldots, w_q) = (u, v, w)$ we suppose given a closed subset S of the polycylinder $\triangle \times \triangle^p \times \triangle^q = \{ (u, v, w) \, \, | \, \, |u|<\epsilon \, , \, \, |v_i| < \epsilon \, , \, \, |w_{\alpha}| < \epsilon \, \}$. Suppose that we let $S_0 = S \cap \{u=0 \}$, and assume the projection $S_0 \rightarrow \triangle^p$ induced by $(0, v, w) \rightarrow v $ is proper.
Then taking a smaller $\epsilon$ if necessary, the projection $S \rightarrow \triangle \times \triangle^p$ induced by $(u, v, w) \rightarrow (u, v)$ will again be proper.
How can I prove last sentence?
Thanks