PROOF 193
Prove.
e arises where the growth rate equals the function itself
On the real numbers ℝ.
A way to read this proof
First separate what is assumed from what is to be proved. At each step, ask which definition or earlier result permits the move. If it is unclear, return to that premise rather than skipping to the conclusion.
Definition D1
For each natural number n, let aₙ = (1 + 1/n)ⁿ.
Proof.
(1)a₁ = 2, a₂ = (3/2)² = 2.25, a₃ = (4/3)³ = 64/27 = 2.370…convention, N
(2)Each aₙ is a rational number determined by finitely many multiplications.convention
∎
Lemma L1
For every n, aₙ < aₙ₊₁
Proof.
(1)Take n copies of 1 + 1/n and one copy of 1, n + 1 positive numbers in all. They are not all equal.convention
(2)The arithmetic mean is (n(1 + 1/n) + 1)/(n + 1) = (n + 2)/(n + 1) = 1 + 1/(n + 1)(1), N
(3)The geometric mean is the (n+1)-th root of ((1 + 1/n)ⁿ · 1), that is, the (n+1)-th root of aₙ.(1), D1, N
(4)They are not all equal, so the inequality is strict: the (n+1)-th root of aₙ < 1 + 1/(n + 1)(2), (3), A1
(5)Both sides are positive. Taking (n+1)-th powers gives aₙ < (1 + 1/(n+1))ⁿ⁺¹ = aₙ₊₁(4), A2, D1
∎
Lemma L2
For every n, aₙ < 3
Proof.
(1)aₙ = (1 + 1/n)ⁿ = Σₖ₌₀ⁿ C(n,k)/nᵏD1, A3
(2)C(n,k)/nᵏ = (1/k!) · (n/n)((n−1)/n)⋯((n−k+1)/n) ≤ 1/k! — each factor is at most 1.(1), N
(3)For k ≥ 1, k! = 1·2·3⋯k ≥ 1·2·2⋯2 = 2ᵏ⁻¹, hence 1/k! ≤ 1/2ᵏ⁻¹N
(4)aₙ ≤ 1 + Σₖ₌₁ⁿ 1/2ᵏ⁻¹(1), (2), (3), A4
(5)Σₖ₌₁ⁿ 1/2ᵏ⁻¹ = 1 + 1/2 + ⋯ + 1/2ⁿ⁻¹ = 2 − 2¹⁻ⁿ < 2A5, N
(6)aₙ < 1 + 2 = 3(4), (5), N
∎
Theorem T1
The sequence (aₙ) converges. Its limit is named e.
Proof.
(1)(aₙ) is increasing.L1
(2)(aₙ) is bounded above by 3.L2
(3)It is a bounded increasing sequence, so a number that (aₙ) approaches exists.(1), (2), A6
(4)Write that number as e. a₁ = 2, the sequence is increasing, and it never exceeds 3, so 2 < e ≤ 3.(3), D1, L1, L2, convention
∎
Statement T2
(eˣ)′ = eˣ. The exponential function eˣ has growth rate equal to itself.
(1)From the definition of the derivative and the exponent law eˣ⁺ʰ = eˣ·eʰ, (eˣ)′ = eˣ · lim (eʰ − 1)/h (h → 0).T1, convention
(2)What remains is lim (eʰ − 1)/h = 1. This follows from the limit definition in T1, but the derivation requires interchanging limits. It is not proved here and is left as a statement.T1
(3)Accepting this statement gives (eˣ)′ = eˣ. The base of the function whose growth rate is itself is exactly this limit e.(1), (2)
Proof omitted.
Remark
Can A6 be removed from this proof?
That aₙ increases without exceeding 3 and that there is a number aₙ approaches are different statements.