= Solution
Put
$$
S_0=0,\qquad S_n=\sum_{i=1}^n\xi_i,
\qquad
N_t=\max\{n\geq0:S_n\leq t\}.
$$
This is the <renewal process> generated by the interarrival times. Since the variables have a density and positive finite mean, $S_n\to\infty$ almost surely, so $N_t$ is finite and tends to infinity with $t$. The strong law gives
$$
\frac{S_n}{n}\longrightarrow\frac1\lambda
\qquad\hbox{almost surely}.
$$
On this probability-one event,
$$
S_{N_t}\leq t<S_{N_t+1}.
$$
Dividing by $N_t$ and using $N_t\to\infty$ gives
$$
\frac{S_{N_t}}{N_t}
\leq\frac{t}{N_t}
<\frac{N_t+1}{N_t}\frac{S_{N_t+1}}{N_t+1}.
$$
Both outside expressions tend to $1/\lambda$, so
$$
\boxed{\frac{N_t}{t}\longrightarrow\lambda\quad\hbox{almost surely}.}
$$
This is the <renewal counting law of large numbers>.
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