Probability that a subsequence of random variables exceeds the maximum

Question

Let $(X_i)_{i=1}^n$ be a sequence of i.i.d. non-negative and bounded random variables. I would like to show that for any $\epsilon>0$ and $k$, one can find an index $\ell$ such that $X_{\ell+j}\geq\max_{i\in\{1,2,\ldots,n\}}X_i-\epsilon$, for all $1\leq j\leq k$, with probability tending to $1$, as $n\to\infty$. In particular, how can we lower bound the probability $$ \Pr\left[\exists\ell:\; X_{\ell+j}\geq\max_{i\in\{1,2,\ldots,n\}}X_i-\epsilon,\;\forall 1\leq j\leq k\right]. $$ Note that for $k=1$ this probability is trivially unity. It seems that standard techniques, such as, the union bound to upper bound the complementary event, are too weak due to the strong dependency of the events indexed by $\ell$.

Answer 1

Hint: The events $\bigcap_{j=1\ldots k} (X_{kl+j} \ge \|X\|_\infty - \epsilon)$ are independent and have the same nonzero probability.