Advanced Calculus (I)

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WEN-CHINGLIEN

Department of Mathematics National Cheng Kung University

WEN-CHINGLIEN Advanced Calculus (I)

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2.4 Cauchy Sequences

Definition

A sequence of points xn ∈ R is said to be Cauchy if and only if for every > 0 there is an N ∈N such that

n, m ≥ N imply |xn− x_m| <

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Definition

A sequence of points xn ∈ R is said to be Cauchy if and only if for every > 0 there is an N ∈N such that

n, m ≥ N imply |xn− x_m| <

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If {xn} is convergent, then {x_n} is Cauchy.

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If {xn} is convergent, then {x_n} is Cauchy.

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Theorem (Cauchy)

Let {xn} ba a sequence of real numbers. Then {xn} is Cauchy if and only if {xn} converges (to some point a in R).

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Theorem (Cauchy)

Let {xn} ba a sequence of real numbers. Then {xn} is Cauchy if and only if {xn} converges (to some point a in R).

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Prove that any real sequence {xn} satisfies

|xn− xn+1| ≤ 1

2ⁿ, n ∈N is convergent.

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Prove that any real sequence {xn} satisfies

|xn− xn+1| ≤ 1

2ⁿ, n ∈N is convergent.

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