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The claim is false it would imply that normal spaces are countably paracompact and this is not the case, see Mary Ellen Rudin, A normal space $X$ for which $X\times I$ is not normal, Fundamenta Mathematicae, 73 (1971/72), 179-186. To show countable paracompactness we use Theorem 5.2.1 in Engelking's General Topology. Let $\{U_n:n\in\omega\}$ be an increasing open cover and $\{V_m:m\in\omega\}$ a cover as in the claim; we then have for every $m$ an $n$ such that $\operatorname{cl}V_m\subseteq U_n$. Define $O_n=\bigcup\{V_m:m\le n:\operatorname{cl}V_m\subseteq U_n\}$; then $\operatorname{cl}O_n\subseteq U_n$ for all $n$ and the $O_n$ form a cover.