Q: Formulate necessary and sufficient conditions for $\alpha_{i} < \beta_{i}$ such that independent (but not identically distributed) Uniform($\alpha_{i}, \beta_{i}$) variables $X_{i}$ converge to $0$.
a) in distribution
b) almost surely
From my workings I think that in both cases we need to have the $\alpha_{i_{s}}$ and $\beta_{i_{s}}$ both converge to $0$ so that the interval $[\alpha_{i},\beta_{i}]$ converges to the singular point $0$. This would imply that the CDF of $X_{i}$ converges to the step function: $0 \quad \text{for} \; x<0$ and $1 \quad \text{for} \; x\geq 0$ and hence $X_{i}$ converges to $0$ in distribution. Moreover, the event $\{ X_{i} \longrightarrow 0 \}$ must have probability $1$ and so we have almost sure convergence.
However this all feels way too simple. Also I would have suspected there to be more relaxed conditions on the parameters for convergence in distribution although I can't imagine any convergence working if the $\alpha_{i}$ and $\beta_{i}$ don't both go to $0$. I don't know if I'm missing something or just overthinking it.
Any help or suggestions would be much appreciated!
P.S. sorry for my slightly wonky LaTeX formatting, I'm still getting to grips with it! :)