# Prove the space of bounded sequences is Banach

http://www.math.ucla.edu/~tao/resource/general/121.1.00s/exam1sol.pdf

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.

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