Mathematical writing

You prove an auxiliary result whose main purpose is to be used in the proof of your main theorem. What is it conventionally called?

A draft says: "If $x^2 = 4$, then $x = 2$." Which revision is a correct, complete statement?

For a function $f : \R \to \R$, which property does this sentence define? "For every $\varepsilon > 0$ there is $\delta > 0$ such that for all $x, y \in \R$ with $|x - y| < \delta$ we have $|f(x) - f(y)| < \varepsilon$."

"Suppose there are finitely many primes $p_1, \dots, p_k$. (1) Let $N = p_1 \cdots p_k + 1$. (2) No $p_i$ divides $N$. (3) Therefore $N$ is prime. (4) But $N$ is larger than every $p_i$, a contradiction." Read as general facts, which step is false?

Your note proves Proposition 4 using an earlier statement labelled "Conjecture 2" (not proved by anyone). How should Proposition 4 be presented?

Which of these sentences follow the lesson's guidelines? (Select all that apply.)

The Lean file records Goldbach's conjecture as def GoldbachConjecture : Prop := ∀ n : ℕ, 2 < n → Even n → ∃ p q : ℕ, p.Prime ∧ q.Prime ∧ p + q = n and separately proves goldbach_up_to_60. Why not write theorem goldbach : ∀ n : ℕ, 2 < n → … := sorry?

Model a text as a list of sentences, each a pair (defines, uses) of sets of symbol names. Write used_before_defined(sentences) returning a list of pairs (i, symbol) — in order of i, and alphabetical within a sentence — for each symbol used in sentence i that is not defined in an earlier sentence (or in sentence i itself).