Fixing a Calculation Mistake on p.51 in "Mathematical Analysis I" by Vladimir A. Zorich?

Question

While reading "Mathematical Analysis I" by Vladimir A. Zorich. I came across this proof regarding the irrationality of $\sqrt{2}$ :

Let $X$ and $Y$ be the sets of positive real numbers such that $\forall x \in X\left(x^{2}<2\right)$ $\forall y \in Y\left(2<y^{2}\right) .$ Since $1 \in X$ and $2 \in Y,$ it follows that $X$ and $Y$ are nonempty sets. Further, since $(x<y) \Leftrightarrow\left(x^{2}<y^{2}\right)$ for positive numbers $x$ and $y,$ every element of $X$ is less than every element of $Y .$ By the completeness axiom there exists $s \in \mathbb{R}$ such that $x \leq s \leq y$ for all $x \in X$ and all $y \in Y$ We shall show that $s^{2}=2$ If $s^{2}<2,$ then, for example, the number $s+\frac{2-s^{2}}{3 s},$ which is larger than $s,$ would have a square less than $2 .$ Indeed, we know that $1 \in X,$ so that $1^{2} \leq s^{2}<2,$ and $0<\Delta:=2-s^{2} \leq 1 .$ It follows that :

$$ \boxed{\left(s+\frac{\Delta}{3 s}\right)^{2}=s^{2}+2 \cdot \frac{\Delta}{3 s}+\left(\frac{\Delta}{3 s}\right)^{2}<s^{2}+3 \cdot \frac{\Delta}{3 s}<s^{2}+3 \cdot \frac{\Delta}{3 s}=s^{2}+\Delta=2} $$

The author made a simple mistake during the expansion as seen in the box above. Therefore, I attempt to fix the proof as follow :

Since $1 \leq s^2$ and $\frac{\Delta}{3}<1$ hold, then : \begin{align*} \left(s+\frac{\Delta}{3s}\right)^2&=s^2+2\cdot\frac{\Delta}{3}+\left(\frac{\Delta}{3s}\right)^2\\ &\leq s^2+2\cdot\frac{\Delta}{3}+\left(\frac{\Delta}{3}\right)^2\\ &<s^2+2\cdot\frac{\Delta}{3}+\frac{\Delta}{3}\\ &= s^2+3\cdot\frac{\Delta}{3}\\ &=s^2+\Delta \\ &= 2 \end{align*}
Hence, $(s+\frac{\Delta}{3s})^2<2$ is true as required.

Are my steps correct in my attempt to fix this part of the proof?

Answer 1

What year was your edition of Zorich published? A copy of the Russian edition from 2012 is here. It has the expansion you write about on the bottom of page number 77 and the calculations there match your correction.

Answer 2

Alternate approach:

Worth looking at the concrete equation $x^2 - 2 = 0$ we are dealing with. We know that there are methods to solve this equation. Among them, there is the Newton approximation (going on the tangent), and the secant method. In this case, since we want values less than the solution, we use the secant method.

So consider the points on the graph $(s, s^2-2)$ below the horizontal, axis, and $(2, 2^2 - 2)$ above the horizontal axis. Join them and consider the point of intersection with the horizontal axis $(s', 0)$. From the picture, we should have $s'> s$, and $s'^2-2 < 0$.

Now, a simple calculation gives $$s' = \frac{2(s+1)}{s+2}$$ Check that \begin{eqnarray} s' - s &=& \frac{2-s^2}{s+2}\\ 2 - s'^2 &=& \frac{2(2-s^2)}{(s+2)^2} \end{eqnarray}

$\bf{Added:}$ Pleased to notice this matches the calculation in Rudin's Principles (exp 1.1). The fog is rising..