A doubt from proof of residue theorem from Tao's notes.
notes 4 Theorem 21 (Residue theorem) Let $U\subset \mathbb{C}$ be a simply connected open set, and let $f: U\setminus S\rightarrow \mathbb{C}$ be holomorphic outside of a closed discrete singular set $S$ (thus all singularities in $S$ are isolated singularities). Let $\gamma$ be a closed curve in $U\setminus S$. Then $$\frac{1}{2\pi i}\int_\gamma f(z) dz=\sum_{z_0\in S} W_{\gamma}(z_0) Res(f;z_0),$$ where only finitely many of the terms on the right-hand side are non-zero
Proof: Being simply connected, $\gamma$ can be contracted to a point inside $U$. This homotopy take values inside some compact subset $K$ of $U$, and thus only contains finitely many of the singularities in $S$. By Rouche's theorem, the winding number $W_{\gamma}(z_0)$ then vanishes for any singularity $z_0$ in $S\setminus K$ (since $\gamma$ can be contracted to a point without touching $z_0$). [...]
So,
Being simply connected, $\gamma$ can be contracted to a point inside $U$
Let $\gamma_1$ be the zero curve (ie $Im(\gamma_1)$ be the point).
doubt 1: can the homotopy invariance (ref below) of winding number be used (as follows)? Since $\gamma_1$ is the zero curve, $\frac{1}{2\pi i}\int_{\gamma_1} \frac{1}{z-z_0} dz =0$. So $W_{\gamma_1}(z_0)=0$. $\gamma$ is homotopic to $\gamma_1$. So, $W_{\gamma}(z_0)=0$.
doubt 2: how is Rouche's theorem (ref below) applicable? There's probably no information about $\mid \gamma_1 (t)-\gamma (t)\mid$ and $\mid \gamma (t)-z_0\mid $
notes 3 lemma 41 (Homotopy invariance) Let $z_0\in \mathbb{C}$, and let $\gamma_0, \gamma_1$ be two closed curves in $\mathbb{C} \setminus \{z_0\}$ and are homotopic as closed curves upto reparameterization in $\mathbb{C} \setminus \{z_0\}$. Then $W_{\gamma_0}(z_0)=W_{\gamma_1}(z_0)$
note: author is referring to Rouche's theorem for winding number and not Rouche's theorem, since the latter is yet to be introduced.
notes 3 corollary 42 (Rouche's theorem for winding number) Let $\gamma_0:[a,b]\rightarrow \mathbb{C}$ be a closed curve, and let $z_0$ lie outside of the image of $\gamma_0$. Let $\gamma_1:[a,b]\rightarrow \mathbb{C}$ be a closed curve such that $$\mid \gamma_1 (t)-\gamma_0 (t)\mid < \mid \gamma_0 (t)-z_0\mid $$ for at $t\in [a,b]$. Then $W_{\gamma_0}(z_0)=W_{\gamma_1}(z_0)$
The notes.
Thanks
