<s>[INST] Mathematical induction problem: $\frac12\cdot \frac34\cdots\frac{2n-1}{2n}<\frac1{\sqrt{2n}}$ This is a problem that I tried to solve and didn't come up with any ideas
.?$$\frac{1}{2}\cdot \frac{3}{4}\cdots\frac{2n-1}{2n}<\frac{1}{\sqrt{2n}}.$$
All I get is $\frac{1}{\sqrt{2n}}\cdot\frac{2n+1}{2n+2}<\frac{1}{\sqrt{2n+2}}$ which evaluates to $1<0$Do you know what to do here ?
 [/INST] This proof true when $\frac{1}{2}\cdot \frac{3}{4}\cdots\frac{2n-1}{2n}<\frac{1}{\sqrt{2n+1}}.$
$$\frac { 1 }{ 2 } \cdot \frac { 3 }{ 4 } \cdots \frac { 2n-1 }{ 2n } \frac { 2n+1 }{ 2n+2 } <\frac { 1 }{ \sqrt { 2n+1 }  } .\frac { 2n+1 }{ 2n+2 } =\frac { 1 }{ \sqrt { 2n+3 }  } .\frac { \sqrt { 2n+3 }  }{ \sqrt { 2n+1 }  } \frac { 2n+1 }{ 2n+2 } =\\ =\frac { 1 }{ \sqrt { 2n+3 }  } \sqrt { \frac { 4{ n }^{ 2 }+8n+3 }{ 4{ n }^{ 2 }+8n+4 }  } <\frac { 1 }{ \sqrt { 2n+3 }  } $$
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