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Sort the Steps: Proving 1/n → 0 Rigorously
publish date: 2026/05/23 22:07:24.565457 UTC
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Sort the following steps into the correct logical order to rigorously prove that \(1/n \to 0\).
Conclude that 1/n < ε for all n > N
Choose N ≥ 1/ε (e.g., N = ⌈1/ε⌉)
Let ε > 0 be any arbitrarily small positive number
Observe that n > N ≥ 1/ε implies 1/n < ε
Correct Answer
(1) Let ε > 0 be any arbitrarily small positive number
(2) Choose N ≥ 1/ε (e.g., N = ⌈1/ε⌉)
(3) Observe that n > N ≥ 1/ε implies 1/n < ε
(4) Conclude that 1/n < ε for all n > N
Explanation
The proof structure: (1) Introduce the challenge — any \(\varepsilon > 0\). (2) Respond — choose \(N \ge 1/\varepsilon\). (3) Verify — \(n > N \Rightarrow n > 1/\varepsilon \Rightarrow 1/n < \varepsilon\). (4) Conclude. This is the \(\varepsilon-N\) argument and is the rigorous definition of a limit at infinity.
Reference
Introduction to Differential Calculus (Systematic Studies with Engineering Applications for Beginners) - 2012