Heine–Borel Theorem
TheoremA subset of Euclidean space $\mathbb{R}^n$ is compact if and only if it is closed and bounded.
Real Analysis
Field: Real Analysis
Sequence of Expressions
Step 6
Step 11
Bolzano–Weierstrass Theorem
TheoremEvery bounded sequence in $\mathbb{R}^n$ has a convergent subsequence.
Step 28
Monotone Convergence Theorem
TheoremIf a sequence of real numbers is monotone and bounded, then the sequence is convergent.
Step 46
Fubini's Theorem
TheoremIf a function is integrable on the product space, the double integral is equal to the iterated integrals.
Step 61
Intermediate Value Theorem
TheoremIf $f$ is a continuous function on $[a, b]$ and $u$ is a value between $f(a)$ and $f(b)$, then there exists a $c \in [a, b]$ such that $f(c) = u$.
Step 96
Extreme Value Theorem
TheoremA continuous real-valued function on a closed and bounded interval attains both a maximum and a minimum value.
Step 132
Heine–Cantor Theorem
TheoremEvery continuous function on a compact set is uniformly continuous.