Proof without induction of the inequalities related to product $\prod\limits_{k=1}^n \frac{2k-1}{2k}$ How do you prove the following without induction:
1)$\prod\limits_{k=1}^n\left(\frac{2k-1}{2k}\right)^{\frac{1}{n}}>\frac{1}{2}$
2)$\prod\limits_{k=1}^n \frac{2k-1}{2k}<\frac{1}{\sqrt{2n+1}}$
3)$\prod\limits_{k=1}^n2k-1<n^n$
I think AM-GM-HM inequality is the way, but am unable to proceed. Any ideas. Thanks beforehand.

Notice that in problem #$1$ if you raise each side to the $n$th power, then it is equivalent to showing that the product of the $n$ factors of the form
$$\left(1-\frac{1}{2k}\right)\tag{1}$$
is greater than $\left(\frac{1}{2}\right)^n$. But that is clearly true since each factor in equation $(1)$ is greater than or equal to $\frac{1}{2}$.
