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I know that if X is Normed linear space then every convergent sequence in X is Cauchy.. is it true the other way around or does it all actually mean one thing ??

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  • $\begingroup$ The definition of a Banach space is a normed linear space in which every Cauchy sequence converges to a limit in the space. A.k.a. a complete normed linear space because it means, equivalently, that the metric $d(x,y)=\|x-y\|$ is a complete metric $\endgroup$ – DanielWainfleet May 3 at 15:19
  • $\begingroup$ An example: Let $X$ be the set of real sequences $(x_n)_{n\in \Bbb N}$ that satisfy $x_n=0$ for all but finitely many $n$. Let $(x_n)_n+(y_n)_n=(x_n+y_n)_n$ and $ r\cdot (x_n)_n=(rx_n)_n$ for $r\in \Bbb R.$ And $\|(x_n)_n\|=\sup_{n\in \Bbb N}|x_n|.$ Then $X$ is a normed linear space.... For $j\in \Bbb N$ let $x(j)=(x_{j,n})_n\in X$ where $x_{j,n}=1/n$ if $n\le j $ and $x_{j,n}=0$ if $n>j.$ Then $(x(j))_{j\in \Bbb N}$ is a Cauchy sequence... But if it converged to $y \in X$ we would have $y=(1/n)_{n\in \Bbb N}\not \in X $. $\endgroup$ – DanielWainfleet May 3 at 15:36
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Every convergent sequence is Cauchy in any normed linear space. This does not imply that $X$ is complete. But the other way holds iff $X$ is complete.

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