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.

Answer for the original question: $\int_0^{1} \frac {e^{x}} {1+x^{n}}dx\to \int_0^{1} e^{x} dx=e-1$ and $\int_1^{a} \frac {e^{x}} {1+x^{n}}dx\to 0$; you can apply DCT for both integrals since the integrand is dominated by $e^{x}$ which is integrable. 
