CONVERGENCE & CAUCHY SEQUENCES : #convergence #cauchy #sequence #complete #metricspace
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One of our main aims in considering metric spaces is to study convergent sequences in a context more general than that of classical analysis. The fruits of this study are many, and among them is the added insight gained into ordinary convergence as it is used in analysis.
Convergent and Cauchy Sequences :
Contents:
1. Convergent sequence and limit point.
2. The limit point of a convergent sequence is unique
3. Cauchy sequence
4. Every convergent sequence in a metric space is a cauchy sequence
5. Complete metric space
1.convergent sequence:-
Let X be a metric space with the metric d and let {xn}= {x1,x2,…,xn,…} be a sequence of points in X.
We say that {xn} is convergent if
(a) ∃ a point x in X such that for each ɛ>0, there exists a positive integer N such that for all n≥N implies d(x,xn)<ɛ
(or)
(b) For each open sphere Sr(x) centered on x, there exists a positive integer N such that xn ꜪSr(x) for all n⩾N.
Here the point x is called the limit point of {xn}.
(or)
(c) d(x,xn) tends to 0 as n tends to ∞
Example:
The sequence {xn} = {1/n} converges to 0.
2. In a metric space, the limit point of a convergent sequence is unique.
Proof :
3. Cauchy sequence:
Let X be a metric space. A sequence {xn} in X is said to be a Cauchy sequence in X for a given ɛ>0, there exists a positive integer N with m, n ⩾N implies d( xn ,xm) < ε.
Example:-
{1/n} is a Cauchy sequence
Result:- Every convergent sequence in a metric space is a cauchy sequence.
Proof :
Complete metric space:-
A complete metric space is a metric space in which every Cauchy sequence is convergent.
i.e., if a Cauchy sequence is convergent means the limit point of the Cauchy sequence must lie in the metric space.
If we see the following example,
Consider X=(0,1] and {1/n} be a Cauchy sequence in (0,1].
We know that {1/n} converges to 0 but 0∉(0,1].
Here {1/n} is a Cauchy sequence but its limit point 0 not lie in (0,1].
∴ (0,1] is not complete.
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#convergence #cauchy #sequence #complete #metricspace
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