I've asked the same question at the Math Stack Exchange site, but I didn't have any luck there. So I'm posting the same question here.

Denote by $r_{s,k}(x)$, the number of ways in which $x$ can be expressed as the sum of $k$ $s^{\text{th}}$ powers of integers. Now, the Hilbert-Waring theorem is equivalent to the following.
$$\forall s\in \mathbb{N}\; \exists\: k \; \: \text{such that} \;r_{s,k}(x)\geq 1\; \forall\; x \in \mathbb{N}$$
Now, call me optimistic, but I've been trying to prove this fact via contradiction. So, assuming the contradiction gives us the following.
$$\forall s\in \mathbb{N}\; \nexists\: k \; \: \text{such that} \;r_{s,k}(x)\geq 1\; \forall\; x \in \mathbb{N}$$
Now, if this is true, then it means that there is *at least one* $s$ for which there exists a corresponding $k$, such that $r_{s,k}(x)\geq 1$. So, for all the other integers $s$, the opposite must hold.

I have come this far till now. If we can show somehow that from the above arguments it must follow that $r_{s,k}(x)=0$ the proof will follow, since for any natural $a$, $r_{s,k}(a^s)$ is always greater than or equal to one.

I'd be grateful to anyone who pitches in any ideas on how to approach this problem, or if it's hopeless to do so. :)

Many thanks in advance!