An inequality for series with fractional exponent

Question

Let $\{a_k\}_{k\in\mathbb{Z}}$ be a bounded and non negative sequence in $\mathbb{R}$, suppose that there exists $N\in\mathbb{Z}$ such that: $a_k=0$, $\forall k\geq N$. Let $p\in(0,1)$. Is true that: $$\biggl( \sum_{k\in\mathbb{Z}}a_k\biggr)^p\leq\sum_{k\in\mathbb{Z}}a_k^p \; ?$$ The series above are convergent. I have tried to use the fact that the function $x^p$, $x\geq 0$, is concave, but this give me: $(x+y)^p\geq 2^{1-p}(x^p+y^p)$, $x\geq0$,bat it is useless. I think that i have to use something inequality for finite sum as above and then pass to the limit. Any help is appreciated.

Answer 1

This follows fom the fact that $$(x+y)^p \leq x^p +y^p $$ which cam be as belov $$(x+y)^p =\frac{x+y}{(x+y)^{1-p} } =\frac{x}{(x+y)^{1-p} } +\frac{y}{(x+y)^{1-p} } \leq \frac{x}{(x)^{1-p} } +\frac{y}{(y)^{1-p} } =x^p +y^p$$ and then by induction $$\left(\sum_{k=l}^n x_k\right)^p \leq \sum_{k=l}^n x_k^p $$ for any finite summation. The infinite case is a consequence of finite summation case.