My question is regarding the question posed at the end of the proof.
My answer is that the result does not hold for all $m \ge 7$ because when $m=7$, the result is $343 \le 128$, which is false.
Is there another way to answer this sort of question besides directly trying different values until you get a false result?
Another question I have regarding this solution is:
In the proof for the inductive step, we start by assuming $k \ge 10$. But along the way, the author mentions $k \ge 1$ and $k \ge 7$ to justify the inequality.
Why do we bother to do this instead of just sticking with $k \ge 10$?