= Solution
Use $\mathbb T^n=\mathbb R^n/(2\pi\mathbb Z)^n$ and the <periodic Sobolev space> convention
$$
\|f\|_{H^s}^2=\sum_{k\in\mathbb Z^n}(1+|k|^2)^s|\widehat f(k)|^2.
$$
The <Cauchy-Schwarz inequality> gives
$$
\sum_k|\widehat f(k)|\leq\|f\|_{H^s}\left(\sum_k(1+|k|^2)^{-s}\right)^{1/2}.
$$
To check the last sum, partition the lattice into $|k|<1$ and dyadic shells $2^j\leq|k|<2^{j+1}$. A shell contains at most $C_n2^{jn}$ points and contributes at most $C_n2^{j(n-2s)}$. This is a convergent <geometric series> exactly when $s>n/2$.
Thus the <Fourier series> $\sum_k\widehat f(k)e^{ik\cdot x}$ converges absolutely and uniformly on the <torus>. Its sum is a <continuous function> and has the original <Fourier coefficients>, so it represents the original periodic <distribution>. In particular the representative is unique and
$$
\boxed{H^s(\mathbb T^n)\hookrightarrow C(\mathbb T^n),\qquad\|f\|_\infty\leq C_{s,n}\|f\|_{H^s}\quad(s>n/2).}
$$
This proves the requested inclusion, with its continuous <Sobolev embedding> estimate.
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