Example for theorem: pointwise but not uniform convergence

Question

I'm reading through Rudin and was trying to understand this theorem- what is a good example to show that this theorem would not hold for pointwise but not uniform convergence?

So, fn that converges to f pointwise, but does not converge to f uniformly for which this theorem fails?

Answer 1

Let $\alpha (x)=x, f_n(x)=n$ of $0<x<\frac 1 n$ and $0$ for $x \geq \frac 1 n$. Let $f(x)=0$ for all $x$. Then $f_n(x) \to f(x)$ for each $x$ but $\int_0^{1}f_n(x)d\alpha (x)=1$ for each $n$ whereas $\int_0^{1}f(x)d\alpha (x)=0$.