= Solution
Let $\alpha$ be a <multi-index> with $|\alpha|\leq k$. The <Cauchy-Schwarz inequality> gives
$$
\int_{\mathbb R^n}|\xi|^{|\alpha|}|\widehat u(\xi)|\,d\xi
\leq\lVert u\rVert_{H^s}
\left(\int_{\mathbb R^n}|\xi|^{2|\alpha|}(1+|\xi|^2)^{-s}\,d\xi\right)^{1/2}.
$$
The second integral is finite precisely under the sufficient inequality $s>|\alpha|+n/2$, which follows from $s>k+n/2$. Hence $(i\xi)^\alpha\widehat u$ is <Lebesgue integrable> for every $|\alpha|\leq k$.
The <Fourier inversion theorem> therefore defines a function
$$
u^*(x)=(2\pi)^{-n}\int_{\mathbb R^n}e^{ix\cdot\xi}\widehat u(\xi)\,d\xi
$$
whose derivatives through order $k$ may be obtained by differentiating under the integral sign. Those derivatives are continuous, so $u^*\in C^k(\mathbb R^n)$. The inverse transform represents the same <tempered distribution> as $u$; consequently $u=u^*$ <almost everywhere>. This is the <Sobolev embedding theorem> $H^s(\mathbb R^n)\hookrightarrow C^k(\mathbb R^n)$ in this range.
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