Prove that if $f'(x)=\frac{1}{x^2+[f(x)]^2},\forall x\in[1,+ \infty[$ then $\lim_{x\to+\infty} f(x)< 1+\frac{\pi}{4} $ Let $f:[1,+ \infty[ \rightarrow \mathbb{R} $, differentiable , $f(1)=1 $ and 
$$f'(x)=\frac{1}{x^2+[f(x)]^2},\forall x\in[1,+ \infty[$$
Prove that, $$\lim_{x\to+\infty} f(x)< 1+\frac{\pi}{4} $$
I managed to prove  $\lim_{x\to+\infty} f(x)\leqslant2 $, but I can't get the above result. Any help would be greatly appreciated. 

Clearly the derivative is bounded by $1/(x^2+1)$, so using the FTC
$$f(x)=1+\int_1^x{dt\over 1+f(t)^2}\le 1+\int_1^x{dt\over 1+t^2}\stackrel{x\to\infty}{\longrightarrow}1+{\pi\over 4}.$$
