Consider the set $$ S = \{ r\in \mathbb{Q} \mid 2^r < 3 \}$$
Our professor wants us to show a few things,
1: that S is bounded from above.
2: if $ 2^\delta < 3 $ there is an $n \in \mathbb{N} $ such that $2^{\delta + 1/n} < 3 $ where $\delta $ is the sup S
3: if $ 2^\delta > 3 $ there is an $n \in \mathbb{N} $ such that $2^{\delta - 1/n} > 3 $
4: Show $2^\delta$ = 3
5: Show there exists a unique solution to $2^p =3$
I'm kind of stuck on where to start at the moment, I felt like using the idea that the set S could be rewritten as $$ S = \{ m/n\in \mathbb{Q} \mid 2^m < 3^n \}$$
could definitely be useful. I feel like if I could figure out the boundedness of the first part I'd be good. From there I'm kinda stumped, if y'all know any good logarithm proofs to look at or any ideas I'd love some help.