What does "If three real numbers $a, x, y$ satisfy the inequalities $a≤x≤a+\frac{y}{n}$ for every integer $n≥1$, then $x=a$" mean? If three real numbers $a, x, y$ satisfy the inequalities $a≤x≤a+\frac{y}{n}$ for every integer $n≥1$, then $x=a$


I saw this theorem in Apostol's Calculus, and even though I understand the proof, I quite don't get what it means, why it's a consequence of the Archimedean property of real numbers (even though I know it's used to prove it), and why it's important to Calculus, as he states. I suppose it has something to do with the inexistence of infinitesimal and infinite numbers that the Archimedean property states, but I still quite don't get it.

Thanks in advance.
The first inequality
says that
$x$ is at least as large as $a$.

If we rewrite the
second inequality as
$x-a
\le \dfrac{y}{n}$,
this says that
the amount $x$ exceeds $a$ by
is less than
$\dfrac{y}{n}$
for any integer $n$.

This is where the
Archimedean axiom comes in.
If the amount $x$ exceeds $a$ by
was greater than zero,
call it $c &gt; 0$,
then
$c \le \dfrac{y}{n}$
for all integers $n$.

This is equivalent to
$n \le yc$.
But Archy says that
for any real
there is a larger integer,
so this can not hold.

Therefore
$x = a$.
You are trying to think too deeply about this.  Subtract $a$ in the inequality and you get $0\leq x-a\leq y/n$ and consider what happens if $x-a&gt;0$. The real number $y/n$ is then positive and becomes less than $x-a$ if $n&gt;y/(x-a)$ and you get a contradiction. Hence $x-a=0$.

Results such as these are trivial and are made complicated only by the fact that they are presented in a book meant for undergraduate students and written using some amount of formalism. If you replace the word "real" in your result with the word "rational" then the statement becomes familiar to any student who has studied fractions and their manipulation. The jump to "real" from "rational" is achieved by noting that all the properties possessed by rational numbers (as far algebraic operations and order relations are concerned) are also possessed by real numbers.