Prove that if a>0, there exists n in N such that 1/n < a < n.
The Attempt at a Solution
I am starting with a>0 and trying to manipulate, algebraically, to get n > a > 1/n.
From a > 0 I can add 1 to both sides to obtain, a+1 > 1. Then I can choose some n such that:
n > a+1 > 1.
From there, divide by n to achieve the desired result for one part of the inequality:
1> (a+1)/n > 1/n. We know from above that n > 1, so I can write:
n > (a+1)/n > 1/n.
From here I'm not sure what to do.