Proofs and formal statements

Tests the sphinx-proof directives: theorem, lemma, corollary, proposition, definition, proof.

Definition

Definition 1

A sequence \(\{x_n\}\) in \(\mathbb{R}\) converges to \(L \in \mathbb{R}\) if for every \(\epsilon > 0\) there exists \(N \in \mathbb{N}\) such that \(|x_n - L| < \epsilon\) for all \(n \ge N\).

Theorem

Theorem 1 (Bolzano–Weierstrass)

Every bounded sequence in \(\mathbb{R}^d\) has a convergent subsequence.

Lemma

Lemma 1

A sequence in \(\mathbb{R}\) is convergent if and only if it is Cauchy.

Corollary

Corollary 1 (of Bolzano–Weierstrass)

Every continuous function \(f : K \to \mathbb{R}\) on a closed bounded subset \(K \subset \mathbb{R}^d\) attains its supremum and infimum.

Proposition

Proposition 1

Expectation is linear: for random variables \(X\), \(Y\) on the same probability space, and constants \(a, b \in \mathbb{R}\), \(\mathbb{E}[aX + bY] = a\mathbb{E}[X] + b\mathbb{E}[Y]\).

Proof

Proof. By Definition 1, it suffices to show that for every \(\epsilon > 0\) there exists \(N\) such that \(|x_n - L| < \epsilon\) for all \(n \ge N\).

Fix \(\epsilon > 0\). Choose \(N\) such that …

The conclusion follows. \(\square\)

Cross-references to numbered statements

By Theorem 1, the sequence has a convergent subsequence. We invoke Lemma 1 to conclude convergence of the whole sequence.

Stacked theorems and proofs

Theorem 2

A first theorem.

Proof. Proof of the first theorem.

Theorem 3

A second theorem, immediately following.

Proof. Proof of the second theorem. Vertical rhythm between consecutive numbered statements should remain consistent.