Sequences (Mathematical Induction)

Can any one help me with this.

We have
$$
U_n=\frac{n^3}{n^4+1}+\frac{n^3}{n^4+2}...+\frac{n^3}{n^4+n}

$$
How to prove that, for all $n$:
$$
\frac{n^4}{n^4+n}\le U_n\le \frac{n^4}{n^4+1}
$$

and what the limit of the sequence ?

I've proved that for $U_0$ but I couldn't prove that for $n+1.$

Thanks too much