Convergence of $(1+x^n)^{1/n}$

Question

I have the sequence of functions $f_n = (1+x^n)^{1/n}$ for $0 \leq x < \infty$.

I can easily see that (as $n$ approaches $\infty$) it is pointwise converging to $1$ for $x\leq 1$ and to $x$ for $x>1$.

I'm trying to figure out whether or not it is converging to these functions uniformly or not, using Dini's theorem I was able to show that it is almost uniformly convergent - but so far have not been able to establish proper uniform convergence, or its non-existence.

My intuition leans towards it not uniformly converging, but I haven't been able to find a proper sequence to contradict it.

Would appreciate any hints.

Thanks a million!

Answer 1

Hint: Show that $f_n(x) - x$ is positive and decreases as $x \rightarrow \infty$, when $n$ is fixed. This will allow you to uniformly bound $|f_n(x) - x|$ in the region $[1, \infty)$.