接下來替日後的行文備好一批標準名詞。以 a 為端點，數線可以伸出四種射線（ray）：open ray {x : x < a} 與 {x : x > a}（不含端點），closed ray {x : x ≤ a} 與 {x : x ≥ a}（含端點）。記號上寫 (−∞, a)、(a, +∞)、(−∞, a]、[a, +∞)——特別提醒：−∞ 與 +∞只是記號，不是 ℝ 的成員，它們的全部意思就是「那一側不設端點」。
CELLS AND INTERVALS
For a ≤ b in ℝ: the open cell(a, b) = {x : a < x < b}; the closed cell[a, b] = {x : a ≤ x ≤ b}; the half-open cells[a, b) and (a, b]; an interval is a ray, a cell, all of ℝ, or ∅ — ten kinds in all; the unit cell is I = [0, 1].
cell（有界區間）四款：open cell (a, b) 兩端不含、closed cell [a, b] 兩端全含、半開半閉各一款；interval（區間）是總稱——四款射線、四款 cell、整條 ℝ、加上空集，共十款。I = [0, 1] 稱 unit cell（單位閉區間），往後各節的常客。
Proof. Write Iₙ = [aₙ, bₙ] with aₙ ≤ bₙ. Nesting gives, for every n, both aₙ ≤ aₙ₊₁ and bₙ₊₁ ≤ bₙ; moreover any left end point stays below any right end point — indeed Iₙ ⊆ I₁ puts every aₙ inside I₁, so the set {aₙ : n ∈ ℕ} is bounded above by b₁. The Supremum Property 6.4 provides ξ = sup{aₙ : n ∈ ℕ}, and aₙ ≤ ξ for every n.
We claim ξ ≤ bₙ for every n. Suppose instead b_m < ξ for some m. Because ξ is the supremum of the left end points, test (ii) of Lemma 6.3 yields an index p with b_m < a_p. Put q = max(m, p). The two monotone chains then give b_q ≤ b_m < a_p ≤ a_q — so the cell I_q = [a_q, b_q] has its left end point beyond its right one, making it void. This contradicts the hypothesis, and therefore aₙ ≤ ξ ≤ bₙ — that is, ξ ∈ Iₙ — for every n.