Calculus, Volume 1, Page 7 Why does n < b^3/(A-b^3/3) lead to contradiction? (Tom M. Apostol 2nd Edition)

Page 7 on "Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra", why does $n < \frac{b^3}{A - \frac{b^3}{3}}$ lead to contradiction?

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Explanation

Because the RHS $\frac{b^3}{A - \frac{b^3}{3}}$ is a fixed value, it does not depend on $n$. But we have $n \geq 1$ mentioned before. When $n \geq \frac{b^3}{A - \frac{b^3}{3}}$, it will have contradiction with the inequality.

Reference

  1. Comments by NickD in the StackExchange Question
MathCalculusTextbook Notes
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