completeness axiom

Completeness axiom of the real numbers $\mathbb{R}$

Every non-empty subset $A\subset \mathbb{R}$ that is bounded from above has a supremum in $\mathbb{R}$, i.e. $\exists T\in \mathbb{R}$ such that $T=\sup A$ (similarily for infimum).
Read: $\mathbb{R}$ has no gaps / you can always reach the boundaries, unlike $\mathbb{Q}$ .

Not every subset of every space has a supremum/infimum in that space

The set $A=\{x\in \mathbb{Q}:x^2\le 2\}$ is bounded, $-\sqrt{2}\le x\le \sqrt{2}$, but $\sup A=\sqrt{2}\notin \mathbb{Q}$, and similarily $\inf A=-\sqrt{2}\notin \mathbb{Q}$.
→ Infimum and Supremum are always defined in a certain space.
We can approach $\sqrt{2}$ abritrarly closely with rational numbers, but never reach it.
But we can define $\sup A=\sqrt{2}$ in $\mathbb{R}$, because $\sqrt{2}\in \mathbb{R}$.

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The Cauchy criterion relies on $\mathbb{R}$ being complete. In $\mathbb{Q}$, a Cauchy sequence of rationals can “converge” to an irrational that isn’t in $\mathbb{Q}$, so Cauchy $\nRightarrow$ convergent there.
This is gives us an equivalent way to define completeness: a space is complete iff every Cauchy sequence converges.

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