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What is meant by cauchy sequence converging in the vector space (for completeness)?

April 26, 2017cauchy completeness converging Mathematics meant sequence vector

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What is meant by cauchy sequence converging in the vector space (for completeness)?

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It is not true that a Cauchy sequence converges to 0! A Cauchy sequence is one whose terms get closer and closer to the other terms as you get farther out in the sequence. A convergent sequence is one whose terms get closer and closer to some fixed vector. Every convergent sequence is Cauchy (if the terms get closer to something, they get close to each other). The issue of completeness is whether every Cauchy sequence is convergent. As an example, consider the continuous functions on [0,1] with the usual inner product defined by =int_0^1 f(x)g(x)dx. Let f_n be a sequence of functions defined as follows: f_n(x)=0 for 0<=x<=1/2 f_n(x)=1 for (1/2)+(1/n)<=x<=1 f_n linear for 1/2<=x<=(1/2)+(1/n). You should verify that the sequence is Cauchy for the norm determined by this inner product. But the sequence does not converge! It is *trying* to converge to the function that is 0 for 0<=x<=1/2 and 1 for 1/2