143 Argument principle

The poles of $T(s)$ are the zeros of $1+R(s)$ together with the poles of $R(s)$, or equivalently the zeros of $1+R(s)$ together with the poles of $1+R(s)$. Let

$\bullet$ $N$ be the number of times that the Nyquist contour of $R$ winds around $-1$, clockwise; let

$\bullet$ $P$ be the number of zeros of $R(s)+1$ in the right half plane;

$\bullet$ $Z$ be the number of poles of $R(s)+1$ in the right half plane;

Then, by the Argument Principle of complex analysis

Here $N=P=0$ by hypothesis, so $Z=0$. Hence $T$ has no poles in the right half plane, hence $T$ is stable.