When random walk is upper unbounded

Consider a random walk $S_n = a_1+\dots+a_n$ where $a_n$ are iid random variables with $Ea_1 = a$ and $E|a_1|<\infty$.

I am interested in the case when $\sup\limits_n S_n>M$ for all $M$ a.s. Can you refer me to a rigorous proof that it not hold for $a<0$?

For $a>0$ it holds by Law of Large Numbers since if $P(\sup\limits_n S_n\leq M) = p$ then either $\lim\limits_n\frac1n S_n$ does not exists, or $\lim\limits_n\frac1n S_n\leq 0$ with probability $p$.

For $a=0$ it holds if $0<Var(a_1)<\infty$ due to the Law of Iterated Logarithm. For $Var(a_1) = 0$ it certainly does not hold.

Solution 1:

When $a<0$, $S_n/n\to a$ almost surely, in particular $S_n\to-\infty$ almost surely, in particular $(S_n)$ is almost surely upper bounded. This implies that $P(\sup_nS_n\ge M)\to0$ when $M\to+\infty$, in particular $P(\sup_nS_n\ge M)<1$ for every $M$ large enough. (Actually, $P(\sup_nS_n\ge M)<1$ for every positive $M$.)

I see no other question in your post.