I try to evaluate this limit:
$$\lim_{n\to+\infty} \frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots(2n)}$$
I considered this inequality
$$\frac{1}{4n}\le\left [ \frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots(2n)} \right]^2\le \frac{1}{2n+1}$$
and so
$$\lim_{n\to+\infty} \frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots(2n)}=0$$
my questions are:
1)- how do I prove the inequality with the principle of induction?
2)- there is another way to solve this limit?