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 
$$
\|X_1+\ldots+X_n\|\leqslant \|X_1+\ldots+X_{n-1}\|+\|X_n\|\leqslant
\|X_1+\ldots+X_{n-1}\|+1.
$$