I encountered an unusual problem and am struggling to prove the following result.
Let $T$ be a positive random variable, and $\{x_i\}$ be a sequence of i.i.d. strictly positive random variables with a finite mean $\mu$ and a finite variance $\sigma^2$. Let $N= \sup_{n} \{S_n \leq T\}$ where $S_n=\sum_{i=1}^n x_i$. I try to prove that: $$\lim_{\mu \rightarrow 0} N\mu \rightarrow T.$$ Intuitively, we are summing up $N$ terms of $x_i$s to get the quantity $T$, and by letting $\mu \rightarrow 0$, I can show that $N \rightarrow \infty$ and $S_n \rightarrow T$ (assuming some boundary conditions hold). And since $S_n$ is approximately $N\mu$ intuitively, the above relationship seems to be true.
I attempt to start from the SLLN for i.i.d. r.v.s: $$\lim_{N \rightarrow \infty} \frac{S_n}{N} \overset{a.s.}{\rightarrow} \mu $$ However $N \rightarrow \infty $ would imply $\mu \rightarrow 0$ and it does not seem to work unless I can multiply both side by $N$.
Any hints on where to start and the mode of convergence are highly appreciated.