A fundamental inequality. I have read in a textbook($p\ge1 ,X_k $ are r.v.):
$$\|\sum_{n=0}^k|X_n|\|_p\le(k+1)\|X_k\|_p\quad\text{where}\mathbb E|X_n|^p\le\mathbb E|X_k|^p  \quad\text{for} \quad 0\le n\le k $$
I tried to extract $|X_k|$ from $(\sum..)^p$ to $\sum(..)^p$ by the "$C_r$-inequality" but I got a extra coefficient "$k^{p-1}$" .Is there any way to circumvent the "$C_r-$inquality"?

(Perhaps there is a typo in the textbook?The author merely want to illustrate the LHS is finite)

Using the triangle inequality, we get 
$$\left\lVert\sum_{n=0}^k|X_n|\right\rVert_p\leqslant \sum_{n=0}^k\lVert X_n\rVert_p\leqslant (k+1)\max_{0\leqslant n\leqslant k}\lVert X_n\rVert_p.$$
The assumption on the expectations is equivalent to $\max_{0\leqslant n\leqslant k}\lVert X_n\rVert_p\leqslant\lVert X_k\rVert_p$, hence the wanted conclusion follows.
