> Since pi over two is not rational or even algrebraic, the integers, > when reduced mod pi over two are an infinite subset of the interval > from 0 to pi over two. it follows that there are integers which when > reduced modulo pi over two, are arbitrarily near pi over two. hence, > there are integers for which tan is arbitrarly large. i hope this > helps.
I'm afraid that argument doesn't work. It is certainly correct to argue that the set of integers, reduced modulo Pi/2, yields an infinite subset of the interval (0, Pi/2). But from this one can conclude only that the resulting subset of (0, Pi/2) has an accumulation point *somewhere* in [0, Pi/2]--and there is no a priori guarantee that this limit point must be Pi/2.