Why we get inequality form an equation?

Question

In the paper linear forms in the logarithms of real algebraic numbers close to 1, it is written on page 5 that-

$\varLambda \leq \frac{1}{by^n}$ (see equation 7 on page 5)

But we get it from an equation. As I understand , it should be $\varLambda = \frac{1}{by^n}$.

How $\varLambda$ could be less than $\frac{1}{by^n}$?

If it is $\varLambda \leq \frac{1}{by^n}$ then why not $\varLambda \geq \frac{1}{by^n}$?

Answer 1

$$a=b$$ implies both $$a\le b$$ and $$a\ge b.$$

It's the author's choice to weaken the comparison for the requirements of the exposition.

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Quiz:

Are these propositions true ? ($\land$ is and, $\lor$ is or)

• $a=b\implies a\le b\land a\ge b$ ?

• $a=b\implies a\le b\lor a\ge b$ ?

• $a=b\implies a< b\lor a=b\lor a> b$ ?

• $a=b\implies a< b\land a=b\land a> b$ ?

Answer 2

From $$\tag6(b+1)x^n-by^n=1$$ we find after dividing by $by^n$ $$\left(1+\frac1b\right)\left(\frac xy\right)^n-1=\frac1{by^n}.$$ This certainly implies (as long as $b,y>0$) that also $$\left|\left(1+\frac1b\right)\left(\frac xy\right)^n-1\right|\le \frac1{by^n}.$$