= Solution
Write $\langle k\rangle=(1+|k|^2)^{1/2}$. There are $O(2^{jn})$ lattice points in the dyadic shell $2^j\leq|m|<2^{j+1}$, while each contributes at most a constant times $2^{-2js}$. The geometric series $\sum_j2^{j(n-2s)}$ converges when $2s>n$, proving $\sum_m\langle m\rangle^{-2s}<\infty$. For $|\alpha|\leq k$ and $u\in H_{s+k}$, the <Cauchy-Schwarz inequality> yields
$$
\sum_m|m^\alpha\widehat u(m)|
\leq\left(\sum_m\langle m\rangle^{-2s}\right)^{1/2}
\left(\sum_m\langle m\rangle^{2s}|m^\alpha|^2|\widehat u(m)|^2\right)^{1/2}
\leq C_s\|u\|_{s+k}.
$$
Thus every derivative <Fourier series> through order $k$ converges absolutely and uniformly. Starting with the continuous sum for $u$, uniform convergence of successive derivative series justifies differentiation, so these are its continuous classical derivatives. \b[$\boxed{H_{s+k}(\mathbb T^n)\hookrightarrow C^k(\mathbb T^n),\ s>n/2}$], with a continuous norm bound.
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