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.