Preliminary: If $N_{\lambda}(t)$ is a random variable representing the number of arrivals in time interval $[0,t]$, for a Poisson point process with a fixed rate $\lambda$, then $$\frac{N_{\lambda}(t)}{t}\xrightarrow{a.s}\lambda$$
Here, $N_{\lambda}(t)$ corresponds to a family of Poisson point process with fixed rate $\lambda$. Note that $N_{\lambda_1}(t)$ and $N_{\lambda_2}(t)$ are independent, for any $\lambda_1 \neq \lambda_2$.
Problem:
Does the following hold? How can we prove it? $$\frac{N_{(1/\sqrt{t})}(t)}{\sqrt{t}}\xrightarrow{a.s}1$$
My attemp: If we assume $S_n$ is the $n^{th}$ time of arrival, then
$$\frac{\sqrt{t}N_{(1/\sqrt{t})}(t)}{S_{N_{(1/\sqrt{t})}(t)+1}}<\frac{N_{(1/\sqrt{t})}(t)}{\sqrt{t}}<\frac{\sqrt{t} N_{(1/\sqrt{t})}(t)}{S_{N_{(1/\sqrt{t})}(t)}}$$
I don't know how to continue from here.