I am reviewing my complex analysis and I got stuck with an exercise about Rouché's theorem. It states: for $0 \leq C \leq \frac{1}{e}$, show that $Ce^z=z$ has exactly one root in the closed unit disc.
For $C < \frac{1}{e}$ I can use easily Rouché's theorem, checking that $z$ and $z-Ce^z$ have the same number of zeroes. Then, for the case $C=\frac{1}{e}$ I don't know what to do. I know the root I want is exactly in 1, so I was thinking about some contour approximating $\lbrace |z|=1 \rbrace$ which excludes 1, but anything I can think about does not give me a contour suitable for applying the theorem.
Am I going in the right direction? Any hint is welcome!