A Multi-Input Multi-Output system with stable transfer matrix $L(s)$ is considered.
The theorem states that if $\rho(L(j\omega)<1$ then the closed-loop system is stable.
In place of the actual theorem its reverse is proved: if the closed-loop is unstable (that is the Nyquist plot of $\det(I+L(s))$ does encircle the origin) then there exists an eigenvalue $\lambda_i(L(j\omega))$ which is larger than $1$ at some $\omega$.
The textbook says: if $\det(I+L(s))$ does encircle the origin then there must exists a gain $\epsilon \in (0, 1]$ and a frequency $\omega'$ such that: $$\det(I+\epsilon L(j\omega'))=0$$
I really cannot figure out the reason why this holds.