# Existence of degree of smooth map between manifold and sphere

Let $$M$$ be an n-dimensional compact, connected, orientable smooth manifold without boundary. Prove that there exists a smooth map $$f:M\rightarrow \mathbb{S}^n$$ such that $$\text{deg}(f) = k$$ for arbitrary $$k\in \mathbb{Z}$$.
Here the degree is defined by $$\int_M f^*\omega = \text{deg}(f)\cdot \int_{\mathbb{S}^n} \omega$$.
HINT: First of all, you can produce maps $$S^n\to S^n$$ of arbitrary degree $$k\in\Bbb Z$$, so it suffices to produce a map from $$M$$ to $$S^n$$ of degree $$1$$. Take a coordinate ball $$B\subset M$$. Using a bump function appropriately to make things smooth, map a sub-ball $$B'\subset B$$ diffeomorphically to $$S^n-\{p\}$$ and $$M-B'$$ to $$p$$.