-- begin header

-- Everything in the header will be hidden in the HTML file.
import data.real.basic

notation `|` x `|` := abs x

meta def ineq_rules : user_attribute :=
{ name := `ineq_rules,
  descr := "lemmas usable to prove inequalities" }

attribute [ineq_rules] add_lt_add le_max_left le_max_right

meta def obvious_ineq := `[linarith <|> apply_rules ineq_rules]
run_cmd add_interactive [`obvious_ineq]
-- end header

# The sandwich theorem

In this demo file, we define limits of sequences of real numbers and prove the sandwich theorem.

/- Definition
A sequence of real numbers $a_n$ tends to $l$ if
$\forall \varepsilon > 0, \exists N, \forall n \geq N, |a_n - l | \leq \varepsilon$.
definition is_limit (a :   ) (l : ) :=
 ε > 0,  N,  n  N, | a n - l | < ε

/- Theorem
If $(a_n)$, $(b_n)$, and $(c_n)$ are three real-valued sequences satisfying $a_n ≤ b_n ≤ c_n$ for all $n$, and if furthermore $a_n→ℓ$ and $c_n→ℓ$, then $b_n→ℓ$.
theorem sandwich (a b c :   )
  (l : ) (ha : is_limit a l) (hc : is_limit c l) 
  (hab :  n, a n  b n) (hbc :  n, b n  c n) : is_limit b l :=
  -- We need to show that for all $ε>0$ there exists $N$ such that $n≥N$ implies $|b_n-ℓ|<ε$. So choose ε > 0.
  intros ε ,
  -- we now need an $N$. As usual it is the max of two other N's, one coming from $(a_n)$ and one from $(c_n)$. Choose $N_a$ and $N_c$ such that $|aₙ - l| < ε$ for $n ≥ Na$ and $|cₙ - l| < ε$ for $n ≥ Nc$.
  cases ha ε  with Na Ha,
  cases hc ε  with Nc Hc,
  -- Now let $N$ be the max of $N_a$ and $N_c$; we claim that this works.
  let N := max Na Nc,
  use N,
  -- Note that N ≥ Na and N ≥ Nc,
  have HNa : Na  N := by obvious_ineq,  
  have HNc : Nc  N := by obvious_ineq,
  -- so for all n ≥ N, 
  intros n Hn,
  -- we have $n≥ N_a$ and $n\geq N_c$, so $aₙ ≤ bₙ ≤ cₙ$, and $|aₙ - l|, |bₙ - l|$ are both less than $\epsilon$.
  have h1 : a n  b n := hab n,
  have h2 : b n  c n := hbc n,
  have h3 : |a n - l| < ε := Ha n (le_trans HNa Hn),
  have h4 : |c n - l| < ε := Hc n (le_trans HNc Hn),
  -- The result now follows easily from these inequalities (as $l-ε<a_n≤b_n≤c_n<l+ε$). 
  revert h3,revert h4,
  unfold abs,unfold max,