Let $(X_n)_{n\geq 1}$ be a sequence of i.i.d variables with Bernoulli distribution $B(p)$. By the strong law of large numbers, we know that $\frac{X_1+\ldots+X_n}{n} \to p$ almost surely.
My question : is there a fixed sequence $(\varepsilon_n)_{n\geq 1}$ tending to zero such that $\Big| \frac{X_1+\ldots+X_n}{n}-p \Big| \leq \varepsilon_n$ for large enough $n$ almost surely, i.e. the event $A=\lbrace \exists n_0, \forall n \geq n_0, \Big| \frac{X_1+\ldots+X_n}{n}-p \Big| \leq \varepsilon_n \rbrace$ has probability one ?
My thoughts : It is not clear to me if $A$ is a tail event, my guess would be that it's not.
Yes. By Hoeffding's inequality $$ \mathbb P\left(\Bigl|\tfrac{X_1+\cdots+X_n}{n}-p\,\Bigr|>\epsilon_n\right)\leq 2\exp\bigl(-2\epsilon_n^2n\bigr). $$ Thus if $\epsilon_n$ is any sequence satisfying $$ \sum_{n\in\mathbb N} \exp\bigl(-2\epsilon_n^2n\bigr)<\infty, $$ it will follow from the Borel-Cantelli lemma that your set $A$ has probability $1$.
In particular we may choose $\epsilon_n=n^{-a}$ for any $a\in(0,\tfrac12)$.
The law of the iterated logarithm gives the optimal growth rate for $\epsilon_n$, namely $$\sqrt{\frac{(2+o(1))\log\log n}{n}},$$ see this comment for an explanation.