Call A a $n \times n$ stochastic matrix and denote with $(\lambda,\textbf{x})$ one of its eigenpair.
Obviously $1$ is an eigenvalue for $A$, indeed follow directly from the definition of row-stochastic [column-stochastic] matrix that $\textbf e$ $=(1\dots1)^{T}$ is a right [left] eigenvector associated with $1$.
Using induced matrix norm $\parallel\parallel_{1}$ or $\parallel\parallel_{\infty}$ , it's easy to prove that the spectral radius $\rho(A)\leq 1$ :
$ |\lambda|= \frac{||A x||}{||x||} \leq max_{||x||=1} ||Ax||= ||A|| $ Now, since $||A||_{1}$ [respectively $||A||_{\infty}$ ] is the maximum absolute column [row] sum of the matrix, we have
$||A||_{1}=1$ if $A$ is a column-stochastic matrix and
$||A||_{\infty}=1$ if $A$ is a row-stochastic matrix,
and then in any case $|\lambda|\leq 1$.