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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?

EDIT:

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 $k \ge 10$?

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2  
Your argument is correct. Having proved that : "if $P(k)$ holds with $k \ge 7$, then it holds for $P(k+1)$" it is not enough to conclude with $P(m)$ holds, for all $m \ge 7$. To do this, we have also to prove that $P(7)$ holds, which is not. –  Mauro ALLEGRANZA Jun 26 at 14:38
    
See also this post: math.stackexchange.com/q/627969/91741 –  Ragnar Jun 26 at 14:38

3 Answers 3

up vote 2 down vote accepted

Even if the case $m=7$ worked, $\textit{this proof}$ would not prove the result for $m=7$ because the base case is at $10$ and we apply induction to prove only for all $m\ge 10$.

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So, even if the case $m=7,8,9$ worked, we still cannot accept that the result works for $m \ge 7$ based on this proof alone? So, in order to be sure, we need to prove the statement "$m^3 \le 2^m$ for $m\ge 7$". Is this what you meant? –  mauna Jun 26 at 14:57
    
@mauna That's exactly what I meant. –  Peter Woolfitt Jun 26 at 14:58

"We only need $k \geq 7$ for the inductive step" is referring strictly to assertion that $k^3 + 7k^2 \leq k^3 + k^3$ within the argument of the inductive step. (See the chain of inequalities.) When $k = 7, \;k^3 + 7k^2 = 7^3 + 7\cdot 7^2 = 7^3 + 7^3$, which is true

Essentially, since the proposition we are aiming to prove, which applies to $k \geq 10$, assures that since $10>7$, we know that the step $k^3 + 7k^2 \leq k^3 + k^3$, because that mini-claim holds for any $k\geq 7$, so that mini-claim certainly holds for $k\geq 10$.

RE: you edit

The author states those claims along the way in the inductive step as a way of stating the minimal requirement at the respective claims for the claim to be true. And since $k\geq 10 \implies k\geq 7$, so any claim that is true for $k \geq 7 is certainly true for $k \geq 10. Likewise, $k \geq 10 \implies k \geq 1$. That's all.

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That $k \ge 7$ is only used to establish the inequality that $ 7k^2 \le k^3$, which is just one step in the proof. That $k \ge 7$ is stated only for clarification: the initial assumption is that $k \ge 10$ which also leads to $ 7k^2 \le k^3$, but is not quite so clear.

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