Exercise 20, § 1

Now prove that, when n increases indefinitely, the polynomial

tends uniformly to −1 in every interval (−1, −ε), and tends uniformly to +1 in every interval (ε, +1), where ε > 0 (notice that $\int_0^1 (1 - t^2)^n\,dt \geq \int_0^1 (1 - t)^n\,dt$). Deduce that the polynomial $g_n(x) = \int_0^x f_n(t)\,dt$ tends uniformly to |x| in (−1, +1), which gives a new proof of the Weierstrass th. (TG, X, p. 37, prop. 3).