What is the weak limit of $f_n \ \mathrm{sign}(f_n - 1)$ if $f_n \to f$ weakly in $L^p([0,1])$?

Question

Let $f_n: [0,1] \to \mathbb R$ be a uniformly bounded sequence in $L^p$. If $f_n \to f$ weakly in $L^p([0,1])$ (up to subsequences), what is the weak limit of the sequence of functions $$g_n = f_n \ \mathrm{sign}(f_n - 1),$$ where sign is the signum function? Can we write it in terms of $f$? Note that $g_n$ is also uniformly bounded in $L^p$, hence it has a weak limit $g$ (up to subsequences). What is the relationship between $g$ and $f$?

Answer 1

The question was edited after In posted this answer.

$(g_n)$ need not converge weakly. [Uniform boundedness only tells you that there exists a subsequence which converges weakly].

For a counter-example take $f_n=1+\frac 1 n$ for $n$ even and $f_n=1-\frac 1 n$ for $n$ odd.