I am studying Stein's Functional Analysis. I worked for a long time but cannot check the following proposition:
Let $f$ be a real valued BMO function, define $f^k$, the truncation of $f$ by
$$f^k(x)= \begin{cases} f(x) &\text{if }|f|<k,\\k &\text{if }f\ge k, \\-k &\text{if }f\le -k\end{cases}$$
Then, $f^k$ is a sequence of BMO functions. The part I cannot solve is $\|f^k\|$ converges to $\|f\|$, where the norm is BMO norm.