Let P(n) be 1/2 * 3/4 ... (2n-1)/2n < 1/sqrt(3n) (note that parentheses around 2n-1 are mandatory). You verified that P(1) is true. The induction step consists of proving that P(k) implies P(k + 1) for all k >= 1. Strictly speaking, this is a true statement just because the conclusion P(k + 1) is true. After all, the overall problem is to prove P(n) for all n. What fails is a particular, most natural way of proving the step. Indeed, the natural way is to say
by the induction hypothesis and then to try proving that the right-hand side is < . However,
has no solutions.
When we strengthen the induction hypothesis, the method above works. Note that the new strict inequality fails for n = 1, so you should either prove a non-strict inequality
for all positive integers n and then say that , or consider n = 2 in the base step.