Uniform Convergence of the Fejer Means of a Continuous Periodic Function on the Torus

We work in the setting of The Flat Torus: Standing Notation, used here with a natural number $n$ satisfying $1\le n$; Euclidean space $\mathbb{R}^{n}$, the periodic class $C_{\mathrm{per}}$ and the absolute value $|t|$ of a real number $t$ are the ones fixed there. For $u\in C_{\mathrm{per}}$ and $N\in\mathbb{N}$ let $\sigma_{N}u$ be the Fejer mean of order $N$ of $u$, and let uniform convergence of a sequence of real-valued maps on $\mathbb{R}^{n}$ be as defined there.

Let $u\in C_{\mathrm{per}}$. (i) Then the sequence $(\sigma_{N}u)_{N\in\mathbb{N}}$ converges uniformly to $u$ on $\mathbb{R}^{n}$. Explicitly, for every real number $\varepsilon$ with $0<\varepsilon$ there is $N_{0}\in\mathbb{N}$ such that

$$|\sigma_{N}u(x)-u(x)|\le\varepsilon\qquad\text{for every }N\in\mathbb{N}\text{ with }N_{0}\le N\text{ and every }x\in\mathbb{R}^{n}.$$