Inspired by Patrick Corn and Myself

This proof is actually lifted verbatim from my longer proof to Two Looming Towers of Power by Patrick Corn. I rely on this lemma heavily while constructing a proof to that question.

We claim that $k=1,l=2$ .

To prove that $a_{n+1} < b_{n}$ , we proceed with induction.

Base case $n=2$ :

$16= a_{3} <b_{2}=27$ is obviously true.

Assuming $a_{n+1} < b_{n}$ is true, we derive

$a_{n+1} \log 2 < a_{n+1} \log 3 < b_{n} \log 3$

$\log 2^{a_{n+1}} <\log 3^{b_{n}}$

$2^{a_{n+1}}<3^{b_{n}}$

$a_{n+2}<b_{n+1}$ as required.

To prove that $b_{n} < a_{n+2}$ , we proceed by induction again.

We will actually prove a stronger condition, that is $4b_{n}<a_{n+2}$

Base case $n=2$ :

$108= 4b_{2}< a_{4}=2^{16}$ is obviously true.

Assuming $4b_{n}<a_{n+2}$ is true,

$4b_{n+1}=4.3^{b_{n}}<4^{b_{n}}.3^{b_{n}}<16^{b_{n}}=2^{4b_{n}}<2^{a_{n+2}}=a_{n+3}$ as required.

The second part of the inequality is proven by Deedlit from MathStackExchange . I could never have applied mathematical induction so skilfully myself!