Another try, now I claim that the inequality holds.
For a random variable $X$, denote $\|X\|=(\mathbb E |X|^\sigma)^{1/\sigma}$. It is indeed a norm.
Then $$ \|X_1+\ldots+X_n\|\leqslant \|X_1+\ldots+X_{n-1}\|+\|X_n\|\leqslant \|X_1+\ldots+X_{n-1}\|+1. $$