<s>[INST] Sequence of functions inequality I need some help to prove that $f_n(x)\leq 1$ , where $f_n:[0,1]\to\mathbb{R},  f_n(x)=\frac{1+nx^2}{(1+x^2)^n}, n\in\mathbb{N}$. 
 I can write $(1+x^2)^n=\binom{0}{n}x^n+\binom{1}{n}x^{n-1}y+\binom{2}{n}x^{n-2}y^2+...+\binom{n-2}{n}x^2y^{n-2}+\binom{n-1}{n}xy^{n-1}+\binom{n}{n}y^n=\\x^n+nx^{n-1}y+\frac{n(n-1)}{2!}x^{n-2}y^2+...+\frac{n(n-1)}{2!}x^2y^{n-2}+nxy^{n-1}+y^n$
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I unsuccessfully tried to find $1+nx^2$ in $(1+x^2)^n$ to show that $1+nx^2\leq(1+x^2)^n$.
 [/INST] $1+nx^2\le(1+x^2)^n$ is true by Bernoulli's inequality, since $x^2\ge0$. Thus $f_n(x)\le1$.
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