u_n(x) converges to u(x) uniformly for all x\in\mathbb{R}.

u'_n(x) converges to 1 for x>0 and to -1 for x < 0, and the convergence is uniform.

u_n(x) is differentiable for all x\in\mathbb{R}.

u(x) is differentiable for all x\in \mathbb{R}.

\lim_{n\to\infty}\int_{-1}^1u_n(t)dt=\int_{-1}^1 u(t)dt.

\int_0^x u_n(t)dt is uniformly converging to \int_0^x u(t)dt for all x\in [0,b]\subset \mathbb{R}.

\lim_{n\to\infty}\int_0^1u_n(t)dt=\int_0^1 u(t)dt.