Here is a proof, but I cannot fully understand why it does not give a proof that $x$ is a bounded sequence (i.e. $x$ is in the space). It seems that the proof only shows that the Cauchy sequence in the space converges to $x$. But $x$ is not shown to be definitely in the space.

  • $\begingroup$ $\|x_n - x\| \to 0$ implies that $x$ is in the space. Try to check it. $\endgroup$ – user99914 Sep 11 '16 at 9:52
  • $\begingroup$ Why is this true? This only implies the limit of xn is x but x may not be in the space. $\endgroup$ – Y.X. Sep 11 '16 at 10:00
  • 2
    $\begingroup$ Use $\|x\| \le \|x_n -x\|+ \|x_n\|$. $\endgroup$ – user99914 Sep 11 '16 at 10:01
  • $\begingroup$ Oh, I see. It is crystal clear now! Many thanks! $\endgroup$ – Y.X. Sep 11 '16 at 10:05

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