Definition D1
lim aₙ = L means the following. For every ε > 0 there exists a natural number N such that |aₙ − L| < ε for every n with n ≥ N.
Proof.
(1)ε is given first, and N is chosen afterward. N may depend on ε.convention
(2)The limit of a function sends x toward c and measures closeness with δ; the limit of a sequence sends n large and measures lateness with N.convention
(3)Hence this definition has no exclusion condition corresponding to 0 < |x − c|. A sequence has no point to skip, and all terms after N are handled at once.convention
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Lemma L1
For natural numbers N₁, N₂, there exists a natural number n with n ≥ N₁ and n ≥ N₂.
Proof.
(1)Let N = max(N₁, N₂). N is N₁ or N₂, so it is a natural number.A13
(2)N ≥ N₁ and N ≥ N₂.(1), A13
(3)n = N is such a natural number.(2)
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Theorem T
If lim aₙ = L₁ and lim aₙ = L₂, then L₁ = L₂.
Proof.
(1)Suppose L₁ ≠ L₂.assumption
(2)L₁ − L₂ ≠ 0, so |L₁ − L₂| ≠ 0, and together with 0 ≤ |L₁ − L₂| this gives 0 < |L₁ − L₂|.(1), A5, A4, A3, A14
(3)Let ε = |L₁ − L₂|/2. Then 0 < ε.(2), A6
(4)For this ε there is a natural number N₁ such that |aₙ − L₁| < ε whenever n ≥ N₁.assumption, (3), D1
(5)For the same ε there is a natural number N₂ such that |aₙ − L₂| < ε whenever n ≥ N₂.assumption, (3), D1
(6)Pick a natural number n with n ≥ N₁ and n ≥ N₂.(4), (5), L1
(7)At that n, |aₙ − L₁| < ε and |aₙ − L₂| < ε.(4), (5), (6)
(8)|L₁ − aₙ| = |−(aₙ − L₁)| = |aₙ − L₁|A12, A2
(9)(L₁ − aₙ) + (aₙ − L₂) = L₁ − L₂A11
(10)|L₁ − L₂| ≤ |L₁ − aₙ| + |aₙ − L₂|(9), A1
(11)|L₁ − aₙ| + |aₙ − L₂| < ε + ε(7), (8), A8
(12)ε + ε = |L₁ − L₂|(3), A7
(13)|L₁ − L₂| < |L₁ − L₂|(10), (11), (12), A9
(14)This cannot hold. Therefore L₁ = L₂.(13), A10
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Corollary C
The symbol lim aₙ refers to a single value.
Proof.
(1)There cannot be two values L satisfying D1.T
(2)If such an L exists, it is unique, so it may be given the name lim aₙ.(1), D1
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