Upper bound on condition number in linear preconditioning

Question

I'm studying iterative methods for solving linear system, and I find the following setting in Wikipedia:
Consider a matrix splitting $A = M-N$, where $A,M,N$ are all symmetric and positive definite matrices. Define iteration matrix $C=I-M^{-1}A$. We aim to show that the condition number $\kappa(I-C)=\kappa(M^{-1}A)$ is not too large, which ensures that iterative method converges fairly fast. Wikipedia gives the following bound on $\kappa(M^{-1}A)$: $$\kappa(M^{-1}A) \leq \frac{1+\rho(C)}{1-\rho(C)}$$ where $\rho(C)$ denotes the spectral radius of matrix $C$. Here we are taking the condition number with respect to the Euclidean 2-norm. That is to say, we have $\kappa(M^{-1}A) = \frac{\sigma_{max}(M^{-1}A)}{\sigma_{min}(M^{-1}A)}$, where $\sigma_{max},\sigma_{min}$ denote the largest/smallest singular value of any matrix, respectively.
Any ideas on proving this upper bound? Any help/hint would be greatly appreciated! (Link for reference: https://en.wikipedia.org/wiki/Preconditioner)