I am completely at a loss as to how to even start Exercise 4, as well as formally defining the two relations in 5. For Exercise 4, I can't figure out what to compare 2^(n-1) to. I mean, there has to be some equality expression somewhere in order to prove something by induction, right? Or am I completely mistaken there? Finally, I already solved whether or not the relations in 5 are equivalence relations, so I'm fine on that front.
I don't want the problems to be solved, just a hint or a clarifying point, or something along those lines.
Edit: I figured 3 out, I was just messing up the induction, as one of the comments pointed out.
For Exercise 3 let $P(n)=\prod_{j=1}^n(1-2^{-j})$ for $n\in \Bbb N.$ If $P(n)\geq 2^{-2}+2^{-n-1}$ then $$P(n+1)-(2^{-2}+2^{-n-2})=$$ $$=(1-2^{-n-1})P(n)-(2^{-2}+2^{-n-2})\geq$$ $$\geq (1-2^{-n-1})(2^{-2}+2^{-n-1})-(2^{-2}+2^{-n-2})=$$ $$=(2^{-2}+2^{-n-1}-2^{-n-3}-2^{-2n-2})-(2^{-2}+2^{-n-2})=$$ $$=((2^{-n-1}-2^{-n-2})-2^{-n-3})-2^{-2n-2}=$$ $$=(2^{-n-2}-2^{-n-3})-2^{-2n-2}=$$ $$=2^{-n-3}-2^{-2n-2}=$$ $$=2^{-2n-2}(2^{n-1}-1)\geq 0.$$