I have seen an unproved claim, which states that given an infinite Bernoulli process with probability $p$ of success, for every $c<p$, the probability that at any given time the success rate is higher than $c$ is positive.
(That is, there exists some $e>0$ such that with that probability there does not exist any $n$ such that the first $n$ trials include at most $c$ successes)
I can't seem to prove it myself, however. If this claim is indeed true, could you help me prove it?
The success rate is higher than $c$ at time $n\geqslant1$ if and only if $S_n\gt nc$, where $S_n$ counts the number of successes during the $n$ first trials. Let $X_n=S_n-nc$, then $(X_n)_n$ is a random walk starting from $X_0=0$ whose steps are $1-c$ with probability $p$ and $-c$ with probability $1-p$. Thus the drift of the random walk is $p-c\gt0$. The law of large numbers says that $X_n\to+\infty$ almost surely, hence the event $A=[\forall n\geqslant1,X_n\gt0]$ has positive probability.
To show that $P[A]\ne0$, assume that $P[A]=0$, then $P[\exists n\geqslant1,X_n\leqslant0]=1$ hence the sequence of stopping times $(T_k)_{k\geqslant0}$ defined by $T_0=0$ and, for every $k\geqslant0$, $T_{k+1}=\inf\{n\geqslant T_k+1\mid X_n\leqslant X_{T_k}\}$, is such that every $T_k$ is almost surely finite. Furthermore, $T_k\to\infty$ almost surely and $X_{T_k}\leqslant0$ for every $k$. In particular, $(X_n)$ does not converge to $+\infty$, which is absurd.
