Show $\exists p \notin f(X) \cup Z$ by Sard

Question

Guillemin & Pallack P83, Ex 2.4.9:

Suppose $X$ compact with $0< \dim(X) < k$ and $f: X \to S^k$. Suppose $Z \subset S^k$ a closed submanifold with $\dim(X) + \dim(Z) = k.$ Show that $I_2(X, Z) = 0.$

The hint says by Sard, $\exists p \notin f(X) \cup Z$. I have read several version of Sards, all saying the image of critical values of a function has measure zero. So I wasn't able to see how can I show $\exists p \notin f(X) \cup Z$?

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Progress following Ted's hint:

$\dim f(X) \cup Z \leq \dim f(X) + \dim Z \leq \dim X + \dim Z = k.$

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To show $\exists p \notin f(X) \cup Z$, that is to show $\exists p \in S^k \setminus (f(X) \cup Z)$.

$\forall p \notin f(X)$ are regular values. So all critical values lies in $f(X)$ and have measure 0.

Answer 1

Hint: When $\dim X<\dim Y$ and $f\colon X \to Y$, what are the regular values of $f$?