Exercise 20, § 1

Show that as $n$ increases indefinitely the polynomial

tends uniformly to $-1$ on every interval $[-1, -\varepsilon]$, and tends uniformly to $+1$ on every interval $[\varepsilon, +1]$, where $\varepsilon > 0$ (note 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 $|x|$ on $[-1, +1]$, which provides a new proof of the Weierstrass theorem (Gen. Top., X, p 313, prop. 3).