I'm attempting to solve Exercise VIII.8.8b in Silverman's Arithmetic of Elliptic Curves:
Prove that $H(x_1+\cdots+x_N)\leq NH(x_1)\cdots H(x_N)$.
This should be elementary, but I'm having trouble figuring out what to do with the $N$ on the right. Plugging in definitions from each end, I imagine the proof should go like
$H(x_1+\cdots+x_N)=H_K([x_1+\cdots+x_N,1])^{1/[K:\mathbb{Q}]}=\prod_{v\in M_K}\max\{|x_1+\cdots+x_N|_v,1\}^{n_v/[K:\mathbb{Q}]}$ $\leq\prod_{v\in M_K}\max\{|x_1|_v+\cdots+|x_N|_v,1\}^{n_v/[K:\mathbb{Q}]}\text{ by the triangle inequality}$ \leq\cdots\text{ (I don't know what goes here) }\cdots\leq $N\prod_{v\in M_K}\left(\prod_{i=1}^N\max\{|x_i|_v,1|\}^{n_v/[K:\mathbb{Q}]}\right)=N\,\,\prod_{i=1}^N\prod_{v\in M_K}\max\{|x_i|_v,1\}^{n_v/[K:\mathbb{Q}]}=NH(x_1)\cdots H(x_n)$
($K$ can be any field containing the $x_i$, e.g. $K=\mathbb{Q}(x_0,\ldots,x_N)$, but this doesn't matter)
I don't see what to do with this $N$ - I can't think of any way to pass it into the product over $v\in M_K$, because that product is infinite, and the only way I can think of to compare these two quantities is to compare term-by-term for each $v\in M_K$. These two ideas for attacking the problem seem to conflict with each other. Can anyone give me a tiny nudge in the right direction? I'm sure I'm missing something obvious.
P.S. I considered induction on $N$, but that requires that I actually prove the result for $N=2$, which I'm unable to do for the above reason.