= Solution
To prove <convergence of positive quadrature on continuous functions>, for a continuous $f$, the <Weierstrass approximation theorem> supplies a <polynomial> $p$ with $\|f-p\|_\infty<\varepsilon$. For every sufficiently large $n$, the <Gaussian quadrature> rule is exact for this fixed <polynomial>. Positivity of its weights and their sum $2$ then give
$$
\left|Q_n(f)-\int_{-1}^1f\right|\leq\sum_jA_j|f(x_j)-p(x_j)|+\int_{-1}^1|f-p|\leq4\varepsilon.
$$
Since $\varepsilon$ is arbitrary, \b[$Q_n(f)\to\int_{-1}^1f$]. Notice that no differentiability of $f$ is needed; uniform <polynomial> approximation and positivity provide the whole argument.
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