1. The problem statement, all variables and given/known data Prove that for any a,b [tex]\in[/tex] R a,b < 0, there is a rational q of the form k/(2n) for n,k[tex]\in[/tex] Z such that a < q < b. 2. Relevant equations None. 3. The attempt at a solution The thing is...I have a solution. Basically, you take the proof for the rationals being dense in R and literally substitude in 2n wherever there is an n. My question is...at the first step for proving the rationals are dense in R, we choose, by the Archimedean Property, a natural number n so that n > max(1/a, 1/(b-a)). So for this proof, we recognize that 2^n > n for all integers n and thus for all natural numbers n. However, in Analysis, "Recognizing" something is true is not enough...and I've got to prove it. And I can prove it..but by induction, which is in the section after this, which leads me to believe that I went about this proof the wrong way. Or do you think the author messed up and included this probem a bit too early in the book? Or is there simply another way to show that 2^n > n for all integers n? If it's necessary to obtain help, I can type up my proof in its full form, but since it literally is (except for the step above that's giving me trouble) the proof for the density of rationals with one variable replaced throughout, I wanted to save myself the trouble. Thanks in advance, this one has been nagging at me for a while.