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.