Uniform convergence of $ x\arctan(nx)$ Let $f_n(x)=x\arctan(nx)$ for $n\geq1$.
Show that $(f_n(x))$ converges uniformly to a function $f$.
In the previous parts I have proved that $f_n(x)$ continuously differentiable and it converges point wise to $f(x)=(\pi/2)|x|$.
How can I find $\sup|f_n(x)-f(x)|$ for all $x\in\mathbb{R}$ in order to prove uniform convergence ?
 A: We will split $\mathbb{R}$ into several pieces. First, we will only consider $[0,\infty)$, since this is a sequence of even functions. Then we prove uniform convergence separately for $[0,1]$ (using Dini's theorem), and then $[1,\infty)$ using the monotonicity of $\arctan(x)$. Establishing uniform convergence over a finite number of sets, guarantees uniform convergence over their union.
Let's begin with $[1,\infty)$.
We know that $\arctan(x) \to \pi/2$ as $x \to \infty$. Moreover, $d/dx \arctan(x) = \frac{1}{1+x^2} > 0$ for all $x$. Thus the convergence is monotonic.
For each $\epsilon >0$, there is a point $r_\epsilon \in \mathbb{R}_{+}$ for which $$|\arctan(x) - \pi/2| < \epsilon$$ for all $x > r_\epsilon$.
Let $N$ be such that $r_\epsilon/N < 1$. Then for all $x \ge 1$ we have $nx > r_\epsilon$ for all $n>N$, which tells us that $$|\arctan(nx) - \pi/2| < \epsilon.$$ Thus for all $x \ge 1$ we have uniform convergence of $x \arctan(nx)$ to $x(\pi/2)$.
For $x \in [0,1]$ we can appeal to Dini's theorem. Dini's theorem says that if a sequence of functions that are continuous, real valued, are pointwise monotonically increasing, and this sequence converges to a continuous function, then the convergence is uniform.
See here: http://en.wikipedia.org/wiki/Dini%27s_theorem
Since $\arctan(x)$ is an increasing function, we have $x \arctan(nx) < x \arctan(mx)$ for $n<m$. Thus the sequence converges pointwise and monotonically.
Thus we know that the sequence converges uniformly on $[0,\infty)$. Now noticing that the sequence is a sequence of even functions, we have uniform convergence on $(-\infty,\infty)$.
If you don't feel comfortable invoking Dini's theorem, you can take the proof of Dini's theorem and specialize it to your own problem.
