Minkowskis inequality implies that $g _k=\sum_{n=1}^{k} |f _{n+1 }-f _n | $ has norm less than $1 $, and there is a hint to use Fatou's lemma to $g _k ^p$.
Then $\int \lim \inf g _k ^p \le \lim \inf \int g _k ^p \le 1$
Now does $\lim \inf g _k = g $? Or can we conclude from the last inequality that $\lim \inf g _k $ must be bounded a.e. and thus equal $g $ a.e.?
As suggested in the comments, the sequence of partial sums $g_k = \sum_{n=1}^k |f_{n+1} - f_n|$ is monotonically increasing and converges to $g$. Then it follows from Monotone Convergence Theorem that $\lim \int g_k = \int g$ and similarly $\lim ||g_k||_p = \int ||g||_p$. Applying triangle inequality and $||f_{n+1} - f_n||_p < 2^{-n}$, you can show each partial sum $||g_k||_p \leq 1$ and then the result follows by passing the inequality to infinite.