Let and consider the set . Prove carefully that a is not bounded above when a>1.
My first thoughts are to use a contradiction, by using the definition of supremum, but I can't seem to find one. Could someone help?
just out of curiosity: how do we know there is such an m? because, to me, this seems like a shell game: we are implcitly assuming that we can make a^m bigger than any real number (in particular, α/a), which is to say, {a^n} is unbounded (circular reasoning).
i mean, it's clear {a^n} is increasing, since a > 1, but increasing without bound? i don't see it in the proof (that is to say, how exactly do we choose m?).
thanks. it is important (from a logical standpoint) that one note that α/a < α (this is what uses the condition that a > 1).
this actually tells us "which" m, it is the first one for which a^m > α/a, that is, the proof becomes constructive.