Consider a random walk $S_n = a_1+\dots+a_n$ where $a_n$ are iid random variables with $Ea_1 = a$ and $E|a_1|<\infty$.
By application of the Law of Large Numbers for $a<0$ we obtain that $S_n/n\to a<0 \Rightarrow S_n\to-\infty \text{ P-a.s.} \quad (1)$ and hence $S_n$ bounded from above with probability one: $ P(\sup\limits_nS_n(\omega) < \infty) = 1\quad (2) $
My thoughts are the following: assume contrary, i.e. $P(\omega:\sup\limits_nS_n(\omega) = \infty) = p>0,\quad (3)$ then for this $\omega$ we have either that limit of $S_n$ does not exist, or that it is $+\infty$, which contradicts with $(1)$.
On the other hand, Didier Piau stated in When random walk is upper unbounded that $ P(\sup\limits_n S_n \geq M)<1 \quad(4) $ for every positive $M$ - and I do not know how to prove it since the way I proved $(2)$ is based on the contradiction with $(3)$, while I cannot justify that $(3)$ contradicts with $(4)$. So my question is how to prove $(4)$?