<s>[INST] Use proof by induction to show that for any positive integer $n\geq 2$ the following holds Use proof by induction to show that for any positive integer $n\geq 2$ the following holds:
$(1+\frac{1}{3})(1+\frac{1}{5})...(1+\frac{1}{2n-1})>\frac{\sqrt{2n+1}}{3}$
Proof: Base case: $n=2$. $LHS:\frac{4}{3}$; $RHS:\frac{\sqrt{5}}{3}$. Clearly, $\frac{4}{3}>\frac{\sqrt{5}}{3}$.
Next, we assume that for some positive integer $n\geq2$ the above statement holds.
Here is where I get consuded. Do I add the next term i.e $(1+\frac{1}{2n})$ to both sides? Or do I multiply the right hand side by that term to continue?
Any tips on how to tackle induction proofs like these? Thanks!
 [/INST] Assuming that $(1+\frac{1}{3})(1+\frac{1}{5})...(1+\frac{1}{2n-1})>\frac{\sqrt{2n+1}}{3}$ consider $(1+\frac{1}{3})(1+\frac{1}{5})...(1+\frac{1}{2n-1})(1+\frac1 {2n+1})$. This quantity is greater than $\frac{\sqrt{2n+1}}{3} (1+\frac1 {2n+1})$. We want to show that  $\frac{\sqrt{2n+1}}{3} (1+\frac1 {2n+1}) >\frac{\sqrt{2n+3}}{3}$. For this use the fact that $\sqrt{2n+1} (1+\frac1 {2n+1})=\sqrt {2n+1} +\frac 1  {\sqrt {2n+1}}$. Puting $x=2n+1$ you will now require the inequality $\sqrt x +\frac  1 {\sqrt x } >\sqrt {x+2}$ or $x+1 >\sqrt {x(x+2)}$. But $x(x+2)=x^{2}+2x < x^{2}+2x+1=(x+1)^{2}$ and you can finish the proof by taking square root on both sides.
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