It many sources it's stated that the winding number is invariant under homotopy, but I've yet to actually see why.
Suppose you have the formal definition of the winding number. So for a continuous loop $\gamma\colon[\alpha,\beta]\to\mathbb{C}\setminus\{a\}$ which doesn't pass through a point $a$, one has the function $\theta(t)=\text{arg}(\gamma(t)-a)\in\mathbb{R}/2\pi\mathbb{Z}$. By the lifting lemma, there exists a continuous $\tilde{\theta}\colon[\alpha,\beta]\to\mathbb{R}$, such that $[\tilde{\theta}(t)]=\theta(t)$, and the winding number of $\gamma$ around $a$ is then defined as $n(\gamma,a)=\frac{\tilde{\theta}(\beta)-\tilde{\theta}(\alpha)}{2\pi}.$
Is there a straightforward proof that the winding number is invariant under homotopy with this definition for continuous loops which do not pass through $a$? Thanks.