<s>[INST] Proving $\left(1+\frac{1}{1^3}\right)\left(1+\frac{1}{2^3}\right)\ldots\left(1+\frac{1}{n^3}\right)<3$ for all positive integers $n$ 
Prove that $\left(1+\dfrac{1}{1^3}\right)\left(1+\dfrac{1}{2^3}\right)\ldots\left(1+\dfrac{1}{n^3}\right)<3$   for all positive integers $n$

This problem is copied from Math Olympiad Treasures by Titu Andreescu and Bogdan Enescu.They start by stating that induction wouldn't directly work here since the right hand side stays constant while the left increases.They get rid of this problem by strengthening the hypothesis.$$\left(1+\dfrac{1}{1^3}\right)\left(1+\dfrac{1}{2^3}\right)\ldots\left(1+\dfrac{1}{n^3}\right)\le3-\dfrac{1}{n}$$
and then proceed by induction.
The problem is that I can't find a motivation for the above change.I mean,we could have subtracted a lot of things from the RHS but what should nudge us to try $\dfrac{1}{n}$?The rest of the proof is quite standard,but I can't see how I am supposed to have thought of it.Is it just experience?Or is it a standard technique?A little guidance and motivation will be appreciated.
 [/INST] If we need to prove the strong inequality and proving the weak inequalities for the simplest to imagine values is easy...
\begin{align*}
\left(3-\frac1n\right)\left(1+\frac1{(n+1)^3}\right)
&=
3+\frac3{(n+1)^3}-\frac1n-\frac1{n(n+1)^3}\\
&=3+\frac{3n-(n+1)^3-1}{n(n+1)^3}\\
&=3+\frac{-n^3-3n^2-2}{n(n+1)^3}\\
&=3+\frac{-1}{n+1}
\frac{n^3+3n^2+2}{n^3+2n^2+n}\\
&\leq3-\frac1{n+1}
\end{align*}
</s>