I need to show that, if $G(k)$ is the solution to Waring's Problem for $k$ and for sufficiently large $n$, then:
$$G(k) \ge k + 1$$
So I need to establish that:
$$x_1^k + x_2^k + \dots + x_k^k = n \tag{1}$$
fails for an arbitrary number of values. I've a few solutions to this, but they seem outside the scope of the course (4th year Ele.Num.Theory).
A hint is that we should use an earlier result that the number of solutions to:
$$x_1 + x_2 + \dots + x_k \le n$$ is $$\binom{n + k}{n}$$
I don't see how I can establish the result given this information. Maybe there's a better way?