<s>[INST] Help with this proof by induction with inequalities. Show that mathematical induction can be used to
prove the stronger inequality $\frac{1}{2}\cdot ...\cdot \frac{2n-1}{2n} < \frac{1}{\sqrt{3n + 1}}$ for all integers greater than 1, which, together with a verification for the case where n = 1, establishes the weaker inequality we originally tried to prove using mathematical induction.
Base Case: P(2)
\begin{aligned}
\frac{1}{2}\cdot\frac{2(2)-1}{2(2)} &< \frac{1}{\sqrt{3(2) + 1}}\\
\frac{3}{8} &< \frac{1}{\sqrt{7}}\\
\frac{1}{8} &< \frac{1}{3\sqrt{7}}\\
\end{aligned}
This is true as $8 > 3\sqrt{7}$.
Inductive Hypothesis: $\frac{1}{2}\cdot ...\cdot \frac{2n-1}{2n} < \frac{1}{\sqrt{3n+1}}$
In the inductive step, we want to show that $\frac{1}{2}\cdot ...\cdot \frac{2n-1}{2n} \cdot \frac{2n+1}{2n+2} < \frac{1}{\sqrt{3n+4}}$.
Using the inductive hypothesis, we can get to the following:
\begin{aligned}
\frac{1}{2}\cdot ...\cdot \frac{2n-1}{2n} \cdot \frac{2n+1}{2n+2} &< \frac{1}{\sqrt{3n+1}}\cdot \frac{2n+1}{2n+2}\\
&< \frac{1}{\sqrt{3n+1}}\cdot 1\\
\end{aligned}
I am not sure how to get to $< \frac{1}{\sqrt{3n+4}}$ from here because i know that    if the denominator would get bigger by adding 3 to it so the inequality wouldn't follow...
 [/INST] Assuming $n\ge0$, we can show that
$$
\frac{2n+1}{2n+2}\le\frac{\sqrt{3n+1}}{\sqrt{3n+4}}\tag1
$$
by squaring both sides to get the equivalent
$$
\frac{4n^2+4n+1}{4n^2+8n+4}\le\frac{3n+1}{3n+4}\tag2
$$
and cross-multiplying to get the equivalent
$$
12n^3+28n^2+19n+4\le12n^3+28n^2+20n+4\tag3
$$
which is true since $n\ge0$.

Therefore, if 
$$
\frac12\frac34\cdots\frac{2n-1}{2n}\le\frac1{\sqrt{3n+1}}\tag4
$$
applying $(1)$, we get
$$
\begin{align}
\frac12\frac34\cdots\frac{2n-1}{2n}\color{#C00}{\frac{2n+1}{2n+2}}
&\le\frac1{\sqrt{3n+1}}\color{#C00}{\frac{\sqrt{3n+1}}{\sqrt{3n+4}}}\\
&=\frac1{\sqrt{3n+4}}\tag5
\end{align}
$$
</s>