<s>[INST] Prove that $\frac{1}{1999} < \prod_{i=1}^{999}{\frac{2i−1}{2i}} < \frac{1}{44}$ 
Prove that $$\dfrac{1}{1999} < \prod_{i=1}^{999}{\dfrac{2i−1}{2i}} < \dfrac{1}{44}$$

from the 1997 Canada National Olympiad.
I have been able to prove the left half of the inequality using induction. Need help with the second part.

Prove $\prod_{i=1}^{n}{(1-\frac{1}{2i})} \ge \frac{1}{2n}$
Define $$p(n)=\prod_{i=1}^{n}{\left(1-\frac{1}{2i}\right)} \tag{1}$$
We wish to show $$p(n)\ge\frac{1}{2n},\quad\forall{n}\in\mathbb{Z^+} \tag{2}$$
For the base case, $n=1$, we have $p(1)=1-\frac{1}{2}=\frac{1}{2}\ge\frac{1}{2}$ which holds.
Assume the induction hypothesis (2) for $n$. Then for $n+1$, we can write:
$$\begin{align}
p(n+1) = \prod_{i=1}^{n+1}{\left(1-\frac{1}{2i}\right)} &= \left(1-\frac{1}{2(n+1)}\right) \cdot \prod_{i=1}^{n}{\left(1-\frac{1}{2i}\right)} \\
&\ge \left(1-\frac{1}{2(n+1)}\right) \cdot \frac{1}{2n} &(\text{by IH}) \\
&\ge \left(\frac{2n+1}{2(n+1)}\right) \cdot \frac{1}{2n}  \\
&\ge \left(\frac{1}{2(n+1)}\right) \cdot \frac{2n+1}{2n}  \\
&> \left(\frac{1}{2(n+1)}\right)  &(\text{for }n\ge1) 
\end{align}$$
This proves the induction step, establishing (2) for all positive $n$. Hence, we conclude
$$\boxed{\prod_{i=1}^{999}{\dfrac{2i−1}{2i}} = p(999) \ge \frac{1}{2\times999} = \frac{1}{1998} > \frac{1}{1999}}$$
Prove $\prod_{i=1}^{999}{\left(1-\frac{1}{2i}\right)} < \frac{1}{44}$
For this part, I have applied AM-GM to get an upper bound with a harmonic sum. 
$$\left(\prod_{i=1}^{999}{\left(1-\frac{1}{2i}\right)}\right)^{1/999} \le \frac{1}{999} \cdot \sum_{i=1}^{999}{\left(1-\frac{1}{2i}\right)} \tag{3}$$
I can't see a way of evaluating the summation on the RHS.  
 [/INST] You have made the problem much hard than it is suppose to be.
For the left hand side, simply observe that $\frac{1}{2}>\frac{1}{3}$ and $\frac{3}{4}>\frac{3}{5}$ and so on... to $\frac{1997}{1998}>\frac{1997}{1999}$. Now, take the product of these inequalities to get the desired result.
For the other inequality, it is a similar trick except you need to observe that $P^2<\frac{1}{1999}<\frac{1}{1936}=\frac{1}{44^2}$.
</s>