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How to prove the space of bounded linear functionals in a linear norm space is complete?

I just have no idea how to use assumptions about $f_n$ to prove $f$ is bounded? I do it like this,

Main idea is since $f_n$ is Cauchy sequence exist N , when $n,m>N$,$||f_n-f_m||<\epsilon$

so $||fn-f||<\epsilon$

$f_n$ is bounded,$||fn||,then $||f||<\epsilon+C_n$ ,also bounded.

Is this right?

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