Show that if $|\gamma_0(t) - y_1(t)| < |\gamma_0(t) - z|$, $w(\Gamma_0, z) = w(\Gamma_1, z)$

I have troubles with the following problem (Amann & Escher Analysis 2):

say $\Gamma_j$ is a piecewise $C^1$ curve parametrized by $\gamma_j \in C(I, \mathbb{C})$ for $j = 0, 1$. Also suppose $z \in \mathbb{C}$ with $|\gamma_0(t) - \gamma_1(t)| < |\gamma_0(t) - z|$ for $t \in I$. Show that $w(\Gamma_0, z) = w(\Gamma_1, z)$, where $w(\Gamma_j, z)$ is defined as $\frac{1}{2\pi i}\int_{\Gamma_j} \frac{1}{\zeta - z} dz$

The hints are: for $\gamma:= \frac{\gamma_1 - z}{\gamma_0 - z}$ show $w([\gamma], 0) = w(\Gamma_1, z) - w(\Gamma_2, z)$ and $|1 - \gamma (t)| < 1$ for $t \in I$.

I have proven the correctness of statements in the hints regarding $\gamma$, and I realize that I need to show something like $|w([\gamma], 0)| < \varepsilon$ for any $\varepsilon > 0$, however, I don't understand how I can make $|1 - \gamma(t)| < 1$ work here. Could someone please help?

If all you're looking for is to prove that $|1 - \gamma(t)| < 1$, then $$\left|1 - \frac{\gamma_1 - z}{\gamma_0 - z}\right| = \left|\frac{(\gamma_0 - z) - (\gamma_1 - z)}{\gamma_0 - z}\right| = \left|\frac{\gamma_0 - \gamma_1}{\gamma_0 - z}\right| < 1$$ is equivalent to $$|\gamma_0 - \gamma_1| < |\gamma_0 - z|,$$ which is the given inequality.

Edit:
If you are instead trying to show that this inequality implies that the winding number of $\gamma$ around $0$ is $0$, then note that this inequality implies that $\gamma$ is contained in the open ball around $1$ with radius $1$. Note that this ball does not contain any point with phase $\pi$ when written in polar coordinates, so it is impossible for $\gamma$ to wind around $0$.