  We are interested in the upper bound of the lifespan of solutions of
semilinear wave equations from above. For the sub-critical case in high
dimensions, it has been believed that the basic tools of its analysis are
Kato's lemma on ordinary differential inequalities and the rescaling argument
in the functional method. But there is a small lack of delicate analysis and no
published paper about this. Here we give a simple alternative proof by means of
improved Kato's lemma without any rescaling argument.
