TITLE: Strong measure zero sets and selection principles
QUESTION [8 upvotes]: A set of reals $X$ is strong measure zero if for any sequence of positive real numbers $ (  \epsilon_n ) _{n \in \omega }$ there is a sequence of open intervals $ (  a_n ) _{n \in \omega }$ which covers $X$ and such that each $ a_i $ has length less than $ \epsilon _i $.
Does anyone know where can I find a proof that the selection principle $\mathcal S_1(\mathcal O,\mathcal O)$ implies Borel property of strong measure zero? I have seen this in The Combinatorics of open covers, but the reference there is to an article which is not in English.
Thank you,

REPLY [2 votes]: To help with @Student's last comment, I post here Baillif's answer.
The proof is very short, actually: given the sequence $\epsilon_i$, let 
$U_i$ be the cover of $X$ by open intervals of length $\epsilon_i$. Then $S_1(O,O)$ ensures that there is a cover of $X$ consisting of (at most) one member of each $U_i$. A more general statement is proved in Theorem 9 in "Finite powers of strong measure zero sets" by M. Scheepers. 
–  Mathieu Baillif