Prove that $\lim_{n\to\infty}\|f_n-f\|_p=\lim_{n\to\infty}\|g_n-g\|_p=0$

Question

I am studying a funtional analysis course and I found the following problem:

Let $(f_n)$ and $(g_n)$ be two sequences such that $f_n\to f$ and $g_n\to g~\mu-a.e$ and assume that there exists some $p\in[1,\infty)$ such that: $$\|f_n-g_n\|_p=\|f_n+g_n\|_p=\|f-g\|_p=\|f+g\|_p=1$$ Show that $\lim_{n\to\infty}\|f_n-f\|_p=\lim_{n\to\infty}\|g_n-g\|_p=0$.

It looks pretty easy, however I cannot find the right path. I suppose I have to use Riesz lemma ($f_n\to f~\mu-a.e$ and $\|f_n\|_p\to\|f\|_p$ implies $\|f_n-f\|_p\to 0 $) but I don't know how to prove that $\|f_n\|_p\to\|f\|_p$. I tried to use triangular inequality, the parallelogram law and to use the definition of norm $p$, but I can't find nothing useful.

Many thanks.

Answer 1

Since $f_n - g_n \to f - g$ $\mu$-a.e. and $\|f_n - g_n\|_p \to \|f - g\|_p$ (by the condition $\|f_n - g_n\|_p = 1 = \|f - g\|$), then $f_n - g_n \xrightarrow{\mathscr{L}^p} f - g$. Similarly, $f_n + g_n \xrightarrow{\mathscr{L}^p} f + g$. Therefore, both $f_n \xrightarrow{\mathscr{L}^p} f$ and $g_n\xrightarrow{\mathscr{L}^p} g$.