In the book "Markov Chains and Stochastic Stability" (page 171, http://probability.ca/MT/) of Meyn and Tweedie there is used the following condition of equicontinuity:
Assume that functions $f_n:\mathbb{R}^d\rightarrow\mathbb {R}$, $n\in\mathbb{N}$ are continuous and they have partial derivative due to each variable. If there exists constant M such that \left\| f_n'(x)\right\|\leq M, for $n\in\mathbb{N}$ and $x\in\mathbb{R}^d$, then the family $\{f_n: n\in\mathbb{N}\}$ is equicontinuous.
Do anybody know how to proof that or where can I find the proof?