= Solution
For every real $s$, the elementary weight estimate
$$
\langle m\rangle^s\leq C_s\langle m-l\rangle^{|s|}\langle l\rangle^s
$$
follows from $\langle x+y\rangle\leq\sqrt2\langle x\rangle\langle y\rangle$, using the reverse rearrangement for negative $s$. If $a$ is smooth its <Fourier coefficients> decay faster than every power, so $c_j=\langle j\rangle^{|s|}|\widehat a(j)|$ is summable. The Fourier convolution formula and <Young convolution inequality> give
$$
\|au\|_s\leq C_s\|c\|_{\ell^1}\|u\|_s.
$$
Initially this holds for trigonometric polynomials, and density extends it to every $u\in H_s$, agreeing with distributional multiplication. Hence smooth multiplication preserves every order.
If $s>n/2$, the stronger estimate $\langle m\rangle^s\leq C_s(\langle l\rangle^s+\langle m-l\rangle^s)$ gives
$$
\|uv\|_s\leq C_s\left(\|u\|_s\|\widehat v\|_{\ell^1}+\|\widehat u\|_{\ell^1}\|v\|_s\right)
\leq C'_s\|u\|_s\|v\|_s,
$$
using the summability estimate from part (b) with $k=0$. Approximation extends the product, and the continuous embedding identifies it with the pointwise product. Completeness from part (a), this estimate and the constant unit make \b[$H_s$ a commutative <Sobolev algebra> for $s>n/2$]. Rescaling the norm by $C'_s$ if necessary gives an equivalent strictly submultiplicative Banach-algebra norm.
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