Why this power inequality for sums of real numbers holds?
For $p > 1$, apply Holder's Inequality to $$\left\lvert \sum_{k=1}^n x_i\right\rvert \leq \sum_{k=1}^n 1\cdot \lvert x_i\rvert \leq n^{1/q} \left(\sum_{k=1}^n \lvert x_i\rvert^p\right)^{1/p} $$ with $q =\frac{p}{p-1}$ is the Holder conjugate of $p$. Now, raise both sides to the power $p$ to get the inequality.
For $p \in (0,1]$, this comes from the following fact:
Lemma. For any sequence $x=(x_1,\dots,x_n)\in\mathbb{R}^n$, $p > 0 \mapsto \lVert{x}\rVert_p$ is non-increasing. In particular, for $0 < p \leq q <\infty$, $$ \left(\sum_i |{x_i}|^q\right)^{1/q} = \lVert{x}\rVert_q \leq \lVert{x}\rVert_p = \left(\sum_i |{x_i}|^p\right)^{1/p}\;. $$ To see why, one can easily prove that if $\lVert{x}\rVert_p = 1$, then $\lVert{x}\rVert_q^q \leq 1$ (bounding each term $|{x_i}|^q \leq |{x_i}|^p$), and therefore $\lVert{x}\rVert_q \leq 1 = \lVert{x}\rVert_p$. Next, for the general case, apply this to $y = x/\lVert{x}\rVert_p$, which has unit $\ell_p$ norm, and conclude by homogeneity of the norm.
By an application of the triangle inequality and the above, taking $q=1$ you get $$ \left\lvert\sum_i {x_i}\right\rvert \leq \sum_i \lvert{x_i}\rvert = \lVert{x}\rVert_1 \leq \lVert{x}\rVert_p = \left(\sum_i |{x_i}|^p\right)^{1/p} $$ and it suffices once again to raise both LHS and RHS to the $p$-th power to conclude.
The first inequality can be shown with induction if we show $$(1+z)^p \leq 1+z^p$$ for $z>1$ and then let $z=x/y$.