Here's a homework problem I have for my class about Discrete Mathematics:
Suppose that we want to prove that
$\frac12\cdot\frac34\cdot\ldots\cdot\frac{2n-1}{2n} < \frac1{\sqrt{3n}}$
for all positive integers $n$.
a) Show that if we try to prove this inequality using mathematical induction, the basis step works but the inductive step fails.
b) Show that mathematical induction can be used to prove the stronger inequality
$\frac12\cdot\frac34\cdot\ldots\cdot\frac{2n-1}{2n} < \frac1{\sqrt{3n + 1}}$
for all integers greater than $1$, which, together with a verification for the case where $n = 1$, establishes the weaker inequality we originally tried to prove using mathematical induction.
I'm not sure how to proceed in the inductive step. I have
$\begin{align*}P(k)&: \frac12\cdot\frac34\cdot\ldots\cdot\frac{2k-1}{2k} < \frac1{\sqrt{3k}}\\ P(k+1)&:\frac12\cdot\frac34\cdot\ldots\cdot\frac{2(k+1)-1}{2(k+1)} < \frac1{\sqrt{3(k+1)}} \end{align*}$
then from that point (which is the very beginning) I'm stumped. This answer exists on Yahoo! Answers, but there's no explanation to the technique. Neither my friends nor my professor have given me a clear step by step answer to the problem. If someone could, it'd be very much appreciated!