Countably Paracompact Lindelöf Space is Paracompact

Theorem
Let $T = \left({X, \vartheta}\right)$ be a Lindelöf space which is also countably paracompact.

Then $T$ is paracompact.

Proof
By the definitions:
 * If $T = \left({X, \vartheta}\right)$ is a Lindelöf space then every open cover of $X$ has a countable subcover.


 * If $T = \left({X, \vartheta}\right)$ is a countably paracompact space then every countable open cover of $X$ has an open refinement which is locally finite.

It follows trivially that every open cover of $X$ has an open refinement which is locally finite.

Hence the result by definition of paracompact.