TITLE: Counterexample around Dini's Theorem
QUESTION [8 upvotes]: "Give an example of an increasing sequence $(f_n)$ of bounded continuous functions from $(0,1]$ to $\mathbb{R}$ which converge pointwise but not uniformly to a bounded continuous function $f$ and explain why Dini's Theorem does not apply in this case"

So clearly Dini's Theorem does not apply, as $(0,1]$ is not a closed interval (or compact metric space), but I can't figure out an example. 
My first thought is $f_n(x)=\frac{1}{x^n}$, but this does not converge pointwise to a bounded continuous function, as $x=1$ is in the interval  
My second thought is $f_n(x)=x^\frac{1}{n}$. This is clearly an increasing sequence of bounded continuous functions (I think?) I believe this converges pointwise to $f(x)=1$ for all $x\in (0,1]$, but I'm struggling to then show why this doesn't converge uniformly to $f(x)=1$
How would I do this? Or is then an easier/better example I could use?

REPLY [4 votes]: Your example $f_n(x) = x^{1/n}$ works fine. What you need is to show that for every $n$ there is an $x_n$ such that $|f_n(x_n)-f(x_n)|$ is greater than some positive constant, for example $x_n = 1/2^n$. That way $f_n$ cannot converge uniformly to $f$.