Is this true? If is, how can I prove this? 
$ (\forall x>1, \ \exists \delta) \ \frac{e^x}{1+x^n}<\frac{e^x}{1+\delta^n}$ 

I use this property to prove for $a>1, \int_0^a \frac{e^x}{1+x^n}\to e-1$ but I’m not sure that why this property is true.

You need $$1<\delta <x$$ 
$$\delta=(1+x)/2$$ works 
