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

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.

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 $$g_n\xrightarrow{\mathscr{L}^p} g$$.