I_3_10 Apostol.

Also introduced notion of "preamble" to share amongst tex docs.

common/Common/Sequence/Arithmetic.tex

@@ -1,13 +1,6 @@
 \documentclass{article}
-\usepackage{amsfonts, amsthm}
-\usepackage{hyperref}
 
-\newtheorem{theorem}{Theorem}
-\newtheorem{custominner}{Theorem}
-\newenvironment{custom}[1]{%
-  \renewcommand\thecustominner{#1}%
-  \custominner
-}{\endcustominner}
+\input{../../preamble}
 
 \begin{document}
 
@@ -21,7 +14,7 @@ $$\sum_{i=0}^n a_i = \frac{(n + 1)(a_0 + a_n)}{2}.$$
 
 \begin{proof}
 
-\href{Arithmetic.lean}{Common.Sequence.Arithmetic.sum\_recursive\_closed}
+\href{Arithmetic.lean}{Common.Sequence.Arithmetic.sum_recursive_closed}
 
 \end{proof}
 

common/Common/Sequence/Geometric.tex

@@ -1,13 +1,6 @@
 \documentclass{article}
-\usepackage{amsfonts, amsthm}
-\usepackage{hyperref}
 
-\newtheorem{theorem}{Theorem}
-\newtheorem{custominner}{Theorem}
-\newenvironment{custom}[1]{%
-  \renewcommand\thecustominner{#1}%
-  \custominner
-}{\endcustominner}
+\input{../../preamble}
 
 \begin{document}
 
@@ -21,7 +14,7 @@ $$\sum_{i=0}^n a_i = \frac{a_0(1 - r^{n+1})}{1 - r}.$$
 
 \begin{proof}
 
-\href{Geometric.lean}{Common.Sequence.Geometric.sum\_recursive\_closed}
+\href{Geometric.lean}{Common.Sequence.Geometric.sum_recursive_closed}
 
 \end{proof}
 

common/preamble.tex

@@ -0,0 +1,12 @@
+\usepackage{amsfonts, amsthm}
+\usepackage{hyperref}
+\usepackage{underscore}
+
+\newtheorem{theorem}{Theorem}
+\newtheorem{xtheoreminner}{Theorem}
+\newenvironment{xtheorem}[1]{%
+  \renewcommand\thextheoreminner{#1}%
+  \xtheoreminner
+}{\endxtheoreminner}
+
+\hypersetup{colorlinks=true, urlcolor=blue}

mathematical-introduction-logic/Enderton/Chapter0.tex

@@ -1,13 +1,6 @@
 \documentclass{article}
-\usepackage{amsfonts, amsthm}
-\usepackage{hyperref}
 
-\newtheorem{theorem}{Theorem}
-\newtheorem{custominner}{Theorem}
-\newenvironment{custom}[1]{%
-  \renewcommand\thecustominner{#1}%
-  \custominner
-}{\endcustominner}
+\input{../../common/preamble}
 
 \begin{document}
 
@@ -20,7 +13,7 @@ Then $x_1 = \langle y_1, \ldots, y_{k+1} \rangle$.
 
 \begin{proof}
 
-\href{Chapter0.lean}{Enderton.Chapter0.lemma\_0a}
+\href{Chapter0.lean}{Enderton.Chapter0.lemma_0a}
 
 \end{proof}
 

one-variable-calculus/Apostol/Chapter_I_3_10.lean

@@ -27,7 +27,7 @@ lemma leq_nat_abs_ceil_self (x : ℝ) : x ≤ Int.natAbs ⌈x⌉ := by
       _ ≤ ↑(Int.natAbs ⌈x⌉) := GE.ge.le h'    
 
 /--
-Theorem 1.29
+Theorem I.29
 
 For every real `x` there exists a positive integer `n` such that `n > x`.
 -/
@@ -40,14 +40,14 @@ theorem exists_pnat_geq_self (x : ℝ) : ∃ n : ℕ+, ↑n > x := by
   exact ⟨x', h⟩
 
 /--
-Theorem 1.30
+Theorem I.30
 
 If `x > 0` and if `y` is an arbitrary real number, there exists a positive
 integer `n` such that `nx > y`.
 
 This is known as the *Archimedean Property of the Reals*.
 -/
-theorem pos_imp_exists_pnat_mul_self_geq {x y : ℝ}
+theorem exists_pnat_mul_self_geq_of_pos {x y : ℝ}
   : x > 0 → ∃ n : ℕ+, n * x > y := by
   intro hx
   let ⟨n, p⟩ := exists_pnat_geq_self (y / x)
@@ -55,4 +55,36 @@ theorem pos_imp_exists_pnat_mul_self_geq {x y : ℝ}
   rw [div_mul, div_self (show x ≠ 0 from LT.lt.ne' hx), div_one] at p'
   exact ⟨n, p'⟩
 
+/--
+Theorem I.31
+
+If three real numbers `a`, `x`, and `y` satisfy the inequalities
+`a ≤ x ≤ a + y / n` for every integer `n ≥ 1`, then `x = a`.
+-/
+theorem forall_pnat_leq_self_leq_frac_iff_eq {x y a : ℝ}
+  : (∀ n : ℕ+, a ≤ x ∧ x ≤ a + (y / n)) → x = a := by
+  intro h
+  match @trichotomous ℝ LT.lt _ x a with
+  | -- x = a
+    Or.inr (Or.inl r) => exact r
+  | -- x < a
+    Or.inl r => 
+    have z : a < a := lt_of_le_of_lt (h 1).left r
+    simp at z
+  | -- x > a
+    Or.inr (Or.inr r) =>
+    let ⟨c, hc⟩ := exists_pos_add_of_lt' r
+    let ⟨n, hn⟩ := @exists_pnat_mul_self_geq_of_pos c y hc.left
+    have hn := mul_lt_mul_of_pos_left hn $
+      have hp : 0 < (↑↑n : ℝ) := by simp
+      show 0 < ((↑↑n)⁻¹ : ℝ) from inv_pos.mpr hp
+    rw [inv_mul_eq_div, ←mul_assoc, mul_comm (n⁻¹ : ℝ), ←one_div, mul_one_div] at hn
+    simp at hn
+    have hn := add_lt_add_left hn a
+    have := calc a + y / ↑↑n
+      _ < a + c := hn
+      _ = x := hc.right
+      _ ≤ a + y / ↑↑n := (h n).right
+    simp at this
+
 end Real

one-variable-calculus/Apostol/Chapter_I_3_10.tex

@@ -1,37 +1,44 @@
 \documentclass{article}
-\usepackage{amsfonts, amsthm}
-\usepackage{hyperref}
 
-\newtheorem{theorem}{Theorem}
-\newtheorem{custominner}{Theorem}
-\newenvironment{custom}[1]{%
-  \renewcommand\thecustominner{#1}%
-  \custominner
-}{\endcustominner}
+\input{../../common/preamble}
 
 \begin{document}
 
-\begin{custom}{1.29}
+\begin{xtheorem}{1.29}
 
 For every real $x$ there exists a positive integer $n$ such that $n > x$.
 
-\end{custom}
+\end{xtheorem}
 
 \begin{proof}
 
-\href{Chapter_I_3_10.lean}{Apostol.Chapter\_I\_3\_10.Real.exists\_pnat\_geq\_self}
+\href{Chapter_I_3_10.lean}{Apostol.Chapter_I_3_10.Real.exists_pnat_geq_self}
 
 \end{proof}
 
-\begin{custom}{1.30}[Archimedean Property of the Reals]
+\begin{xtheorem}{1.30}[Archimedean Property of the Reals]
 
 If $x > 0$ and if $y$ is an arbitrary real number, there exists a positive integer $n$ such that $nx > y$.
 
-\end{custom}
+\end{xtheorem}
 
 \begin{proof}
 
-\href{Chapter_I_3_10.lean}{Apostol.Chapter\_I\_3\_10.Real.pos\_imp\_exists\_pnat\_mul\_self\_geq}
+\href{Chapter_I_3_10.lean}{Apostol.Chapter_I_3_10.Real.exists_pnat_mul_self_geq_of_pos}
+
+\end{proof}
+
+\begin{xtheorem}{1.31}
+
+If three real numbers $a$, $x$, and $y$ satisfy the inequalities
+$$a \leq x \leq a + \frac{y}{n}$$
+for every integer $n \geq 1$, then $x = a$.
+
+\end{xtheorem}
+
+\begin{proof}
+
+\href{Chapter_I_3_10.lean}{Apostol.Chapter_I_3_10.Real.forall_pnat_leq_self_leq_frac_iff_eq}
 
 \end{proof}