<s>[INST] Show that $\left(1+\frac{1}{1^3}\right)\left(1+\frac{1}{2^3}\right)\left(1+\frac{1}{3^3}\right)\cdots\left(1+\frac{1}{n^3}\right) < 3$ I have this problem which says that for any positive integer $n$, $n \neq 0$ the following inequality is true: $$\left(1+\frac{1}{1^3}\right)\left(1+\frac{1}{2^3}\right)\left(1+\frac{1}{3^3}\right)\cdots\left(1+\frac{1}{n^3}\right) < 3$$
This problem was given to me in a lecture about induction but any kind of solution would be nice.And also I'm in 10th grade :)
 [/INST] Claim:

For all positive integers $n$, we have
$$\prod_{k=1}^n \left(1+\frac{1}{k^3}\right) < e$$
or equivalently
$$\sum_{k=1}^n \ln\left(1+\frac{1}{k^3}\right) < 1$$
Proof:

It suffices to show that
$$\sum_{k=1}^n \ln\left(1+\frac{1}{k^3}\right) < 1-\frac{1}{(n+1)^2}\tag{*}$$
holds for all positive integers $n$.

To prove $(*)$, proceed by induction on $n$.

By direct evaluation, $(*)$ holds for the base case $n=1$.

Suppose $(*)$ holds for some positive integer $n$.
\begin{align*}
\text{Then}\;\;&\sum_{k=1}^{n+1} \ln\left(1+\frac{1}{k^3}\right)\\[4pt]
=\;&
\left(
\sum_{k=1}^n \ln\left(1+\frac{1}{k^3}\right)
\right)
+
\ln\left(1+\frac{1}{(n+1)^3}\right)
\\[4pt]
<\;&
\left( 1-\frac{1}{(n+1)^2}\right)
+
\ln\left(1+\frac{1}{(n+1)^3}\right)
&&\text{[by the inductive hypothesis]}
\\[4pt]
<\;&
\left( 1-\frac{1}{(n+1)^2}\right)
+
\frac{1}{(n+1)^3}
&&\text{[since $\ln(1+x) < x$, for all $x > 0$]}
\\[4pt]
=\;&\left(1-\frac{1}{(n+2)^2}\right)-\frac{n^2+n-1}{(n+1)^3(n+2)^2}
\\[4pt]
<\;&1-\frac{1}{(n+2)^2}
&&\text{[since $n^2+n-1 > 0$]}
\\[4pt]
\end{align*}
which completes the induction, and thus proves the claim.
</s>