You're using the Heine-Borel theorem, which assumes the standard topology on
; that is, the topology generated by
To see that
compact in our new topology (which is often called the lower-limit topology), we must find an open cover of
which has no finite refinement. I believe the cover
will do the trick.
(For what it's worth, the right endpoint of the interval
really doesn't matter, so long as it's greater than 1. We just need an open set whose intersection with