Uniform Convergence of \{\frac{n^2x}{1+n^3x}\} on Different Intervals

[alligatorman]

I need to determine whether the sequence $$\{\frac{n^2x}{1+n^3x}\}$$ is uniformly convergent on the intervals:

[1,2]
[a,inf), a>0

For the first one, I notoced the function is decreasing on the interval, so the $$\sup|\frac{n^2x}{1+n^3x}|$$ will be when x=1, and when x=1, the sequence goes to 0, proving uniform convergence.

I'm not so sure how to approach the second one, because the sequence may not necessarily be decreasing on [a,inf)

Any help?

 

[mathman]

The general idea is that for a>0, there will be an N such that the sup of the sequence will be at x=a for n>N.