Square-section hammer-over strip, butted up against the threshold. It is
available in colors around here- go for a brown that is close to threshold
color, and avoid the silver or gold. Don't use tack strip in the door
opening. You could also cut a kerf in the threshold strip, but that would be
nasty to the next owner who wants to go back to wood floors. Don't carpet
the threshold strip- it'll wear through in months.
In some cases like this you would use a tack strip at the door opening
and tuck the carpet between the back of the strip and the threshold. If
the surface is flat at the threshold then you would use a metal tack
threshold strip. You would stretch the carpet up to it, cut it, and
then hammer the top down.
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