I do not understand this particular step in a proof using the Well Ordering Principle Below is a proof using the Well Ordering Principle. I get lost starting at $(13)$...
$$
\begin{aligned}
P(c): &:=c^{3} \leq 3^{c} \\
& \equiv c^{3} \leq 3(c-1)^{3} \\
& \equiv c \leq \sqrt[3]{3} \times(c-1)
\end{aligned}
$$
I don't understand how we get $c^{3} \leq 3(c-1)^{3}$ from $c^{3} \leq 3^{c}$.. why does the righthand side of the inequality change like that?


You want to show that
$c^3 \le 3^c$
by induction.
This means that you want to show that
$(c-1)^3 \le 3^{c-1}$
implies that
$c^3 \le 3^c$.
By the induction hypothesis,
$3^c = 3\cdot 3^{c-1}
\ge 3(c-1)^3
$.
Therefore,
if you can show that
$3(c-1)^3
\ge c^3
$
then
$3^c \ge c^3
$.
Now we can proceed as above.
Taking cube roots,
$3(c-1)^3
\ge c^3
$
becomes
$3^{1/3}(c-1)
\ge c
$
or
$c(3^{1/3}-1)
\ge 3^{1/3}
$
or
$c
\ge \dfrac{3^{1/3}}{3^{1/3}-1}
= \dfrac{1}{1-3^{-1/3}}
\approx 3.26...
$
so the induction will work for
$c \ge 4$.
Since
$4^3 (64) < 3^4 (81)
$,
$c^3 \le 3^c$
for $c \ge 4$.
(With induction,
you always have to check
the base case.)
