Hurewicz's reformulation of Menger property Many papers on selection principles and also the Wikipedia article mention the reformulation of Menger's property given by Hurewicz.
Menger's formulation for metric spaces was: Every basis $\mathcal B$ of the topology of $(X,d)$ contains a sequence $(B_n)$ such that $\operatorname{diam}B_n\to 0$ and $\{B_n; n=1,2,\dots\}$ covers $X$.
Hurewicz has shown that this is equivalent to: For every sequence $\mathcal U_n$ of open covers of $X$ there exists a sequence of finite subsets $\mathcal F_n\subseteq \mathcal U_n$ such that $\bigcup_{n=1}^\infty \mathcal F_n$ covers $X$.
As the reference for the proof that these two formulations are equivalent (in metric spaces) I only found the original Hurewicz's paper: 
Hurewicz, Witold (1926). "Über eine verallgemeinerung des Borelschen Theorems". Mathematische Zeitschrift. 24.1: 401–421. DOI: 10.1007/BF01216792, eudml, GDZ.
Are you aware of some English text which contains proof of this result? (Or, if the proof is simple enough, can you outline it here? But this does not seem very likely - I guess after briefly looking through Hurewicz's paper.)

Please see Theorem 2 in the following paper:  
On some properties of Hurewicz, Menger, and Rothberger,
authors Arnold Miller, David Fremlin,
Fundamenta Mathematicae (1988)
Volume: 129, Issue: 1, page 17-33
ISSN: 0016-2736 
available online at
https://eudml.org/doc/211668 
