- #1
sonofagun
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I'm reading the proof that a cauchy sequence is convergent.
Let an be a cauchy sequence and let ε=1. Then ∃N∈ℕ such that for all m, n≥N we have
an-am<1. Hence, for all n≥N we have an-aN<1 which implies an<aN+1. Therefore, the set {n∈ℕ: an≤aN+1} is infinite and thus {x∈ℝ : {n∈ℕ: an≤x} is infinite} ≠ ∅.
I can't make sense of the last set. What does it represent and why is it not empty?
Let an be a cauchy sequence and let ε=1. Then ∃N∈ℕ such that for all m, n≥N we have
an-am<1. Hence, for all n≥N we have an-aN<1 which implies an<aN+1. Therefore, the set {n∈ℕ: an≤aN+1} is infinite and thus {x∈ℝ : {n∈ℕ: an≤x} is infinite} ≠ ∅.
I can't make sense of the last set. What does it represent and why is it not empty?