Question
Prove that every convergent sequence is Cauchy.
Answer :
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Let ((x_n)) be a convergent sequence in a metric space ((X,d)) and suppose (x_n \to x) for some (x \in X). By definition of convergence, for every (\varepsilon > 0) there exists a natural number (N) such that (d(x_n, x) < \varepsilon/2) for all (n \ge N).
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Related Questions
जाँच कीजिए कि पूर्णांकों का समुच्चय गणनीय है या नहीं।
Disprove the statement:
by providing a suitable counter-example.
श्रेणी के अभिसरण की जाँच कीजिए।
जाँच कीजिए कि अनुक्रम , जिसे इस प्रकार से परिभाषित किया जाता है:
, और
.
अभिसारी है या नहीं।
Test the convergence of the series: .
श्रेणी , की अभिसरण त्रिज्या ज्ञात कीजिए। जहाँ
है।
Discuss the convergence of the p-series by using the integral test.