Let the -th row of be .
Then the -th element in the -th row of is:
Now, as is upper trianglar:
and if this is zero and as by definition for
Now suppose that for some for all
because is upper triangular, and:
because by hypothesis , for all .
Which completes a proof by induction that for all , which proves that is upper triangular.
(it is easier to see what is going on doing this element by element along the a row of )