([KRD10, 6.4.N]) If \(f\) and \(g\) are bounded on \([a,b]\) and both are Riemann integrable on \([a,b]\), show that \(fg\in\mathcal R\) on \([a,b]\).
Solution
1. Reduce to squares and sums.Proof: the identity \(fg = \frac14\big((f+g)^2 - (f-g)^2\big)\) reduces the claim to: \(\mathcal{R}\) is closed under addition (and scalar multiplication), and \(h \in \mathcal{R} \implies h^2 \in \mathcal{R}\).
2. \(\mathcal{R}\) is a vector space.Proof: for \(f, g \in \mathcal{R}\) and a partition \(P\), the oscillation bound \(\mathrm{osc}_{[x_{i-1},x_i]}(f+g) \le \mathrm{osc}_{[x_{i-1},x_i]}(f) + \mathrm{osc}_{[x_{i-1},x_i]}(g)\) gives \(U_{f+g}(P) - L_{f+g}(P) \le \big(U_f(P) - L_f(P)\big) + \big(U_g(P) - L_g(P)\big)\), which tends to \(0\) along a suitable sequence of partitions.For a scalar \(c\), the oscillation bound \(\mathrm{osc}(cf) = |c|\,\mathrm{osc}(f)\) gives \(U_{cf}(P) - L_{cf}(P) = |c|\big(U_f(P) - L_f(P)\big) \to 0\) along the same sequence of partitions.Hence \(f+g, cf \in \mathcal{R}\).
3. \(h \in \mathcal{R}\) bounded implies \(h^2 \in \mathcal{R}\).Proof: let \(|h| \le M\).For \(u, v\) in an interval, \(|h(u)^2 - h(v)^2| = |h(u) - h(v)||h(u) + h(v)| \le 2M|h(u) - h(v)|\), so \(\mathrm{osc}(h^2) \le 2M\,\mathrm{osc}(h)\) on every subinterval; hence \(U_{h^2}(P) - L_{h^2}(P) \le 2M\big(U_h(P) - L_h(P)\big) \to 0\).So \(h^2 \in \mathcal{R}\).
4. Conclude.Proof: \(f+g, f-g \in \mathcal{R}\) (<1>2), so \((f+g)^2, (f-g)^2 \in \mathcal{R}\) (<1>3), and \(fg = \frac14[(f+g)^2 - (f-g)^2] \in \mathcal{R}\).
5. Q.E.D.