All of analysis rests on a single property: the completeness of ℝ. Remove it, and its theorems fall one by one.

Real analysis

What analysis studies

Real analysis studies functions of a real variable by means of a single tool: passing to the limit. Continuity, derivative, integral, series — everything reduces to it.

It is not an assembly of independent results. It is an edifice resting on one property, and it collapses entirely if that property is removed.

The single foundation

That property is completeness: every non-empty subset of $\mathbb{R}$ that is bounded above has a least upper bound.

The great theorems of analysis are not merely deduced from it — they are equivalent to it. Granting any one of them amounts to granting completeness, and conversely.

Theorem	Statement
Intermediate value	A continuous function that changes sign has a root
Extreme value	A continuous function on a closed interval attains its maximum
Bolzano–Weierstrass	Every bounded sequence has a convergent subsequence
Cauchy criterion	Every Cauchy sequence converges

The edifice over ℚ

The best way to see what completeness buys is to take it away. Let us work over $\mathbb{Q}$ , where it fails, and take:

$f(x) = x^2 - 2 \quad \text{on} \quad [1, 2] \cap \mathbb{Q}$

This function is continuous, $f(1) = -1 < 0$ and $f(2) = 2 > 0$ . The intermediate value theorem would require it to have a root. It has none: its root is $\sqrt{2}$ , which is not rational.

The same gap brings down the other three. The sequence of decimal approximations of $\sqrt{2}$ is Cauchy in $\mathbb{Q}$ and does not converge there; it is bounded and has no convergent subsequence there.

ℝ

√2
1
2

Derivative and integral are limits

The two central operations are nothing but disguised passages to the limit:

$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

The Riemann integral is likewise the limit of sums of rectangle areas whose width tends to zero.

Neither makes sense without the guarantee that these limits exist. That is why differential and integral calculus could not have been founded on the rationals alone.

What "rigorous" came to mean

Analysis long ran on intuitions of continuity and of the infinitely small that gave correct results without anyone knowing why.

The $\varepsilon$ – $\delta$ formulation and the explicit construction of $\mathbb{R}$ replaced those intuitions with checkable statements. The gain is not in finding new results: it is in being able to say exactly which hypothesis each theorem depends on.

Summary

Remove completeness and analysis stops: its theorems are equivalent to it, not merely subordinate to it.