As a step in proving that the union of intervals $B_E \subset [0,1]$ (where $E$ is the set of infinite Bernoulli sequences, e.g. 0.0110...., such that some finite pattern repeats infinitely often) is a Borel set, the authors in this book build the set as follows. First, they define $E_n$ as the set of sequences where the pattern occurs beginning at position $n$ and $B_{E_n}$ to be the corresponding union of intervals. Then they state without proof that \begin{align} B_E = \bigcap^\infty_{k=1} \; \bigcup_{n \geq k} \; B_{E_n} \end{align}
After thinking about this statement for a few days, I have now convinced myself that this does indeed represent the set of Bernoulli sequences (and the corresponding intervals) where a given finite pattern repeats infinitely often.
On the other hand, if someone asked me to prove that this infinite intersection of infinite unions does represent $B_E$, I'm not sure where I should start (I think I could do a hand waving argument, though). So my question is, how do you formally prove equivalence between the LHS and RHS?