= Solution
Let $K$ be a <number field> of degree $d$, and normalize its <absolute values> so that the <product formula> is $\prod_v|a|_v^{n_v}=1$ for $a\ne0$, where $n_v=[K_v:\mathbb Q_v]$; at a complex place use the ordinary modulus with weight two. For $P=[a_0:\cdots:a_N]$, define
$$
\boxed{H(P)=\prod_{v\in M_K}\max_i|a_i|_v^{n_v/d},\qquad h(P)=\log H(P).}
$$
Only finitely many factors differ from one. Rescaling all coordinates multiplies the local maxima by $|a|_v$, whose weighted product is one, so the <projective height> is well-defined. It is at least one by comparison with any nonzero coordinate. On enlarging $K$ to $L$, the identity $\sum_{w\mid v}[L_w:\mathbb Q_v]=[L:K][K_v:\mathbb Q_v]$ makes the normalized product unchanged. Thus this is an absolute height on algebraic projective points. For coprime rational integer coordinates it is simply their maximum absolute value.
For a projective variety $V/K$ and an embedding $i:V\hookrightarrow\mathbb P^N$, take $h_i=h\circ i$. Changing the basis of the same projective linear system changes the height by a bounded amount: the coordinate transformation and its inverse have fixed local coefficient bounds, equal to one outside finitely many places. If $L$ is ample and $L^m$ is very ample, set $h_L=m^{-1}h_i$ for an embedding from $L^m$, up to a bounded function. A general line bundle can be written $(L\otimes A)\otimes A^{-1}$ with both $A$ and $L\otimes A$ very ample, defining its height as the difference of their embedding heights.
This is the <Weil height machine>, with
$$
h_{L\otimes M}=h_L+h_M+O(1),\qquad h_{\phi^*L}=h_L\circ\phi+O(1).
$$
The first relation follows from the Segre embedding, whose local maximum is the product of the two maxima; tensor powers similarly use the Veronese embedding. The second follows by pulling back the defining sections. These constructions make the choices equivalent modulo bounded functions. For a fixed homogeneous morphism with no base point, upper local height bounds come from its coefficients, and lower bounds from the <Projective Nullstellensatz> expressing coordinate powers in the ideal of its defining forms. The constants are one outside a finite set of places. This gives the uniform bounded errors and shows why heights belong to line-bundle classes rather than being independent of every projective embedding.
The <Northcott lemma> states that for fixed $B$ and $D$, the algebraic points of $\mathbb P^N$ with $H(P)\leq B$ and $[\mathbb Q(P):\mathbb Q]\leq D$ form a finite set. In particular the $K$-rational points of bounded projective height are finite. Here is a proof, including the degree bound.
First let $\alpha$ have degree $e\leq D$ and $H(\alpha)=H([1:\alpha])\leq B$. Write its primitive minimal polynomial as $a_e\prod_{j=1}^e(T-\alpha_j)$, with positive integer $a_e$. The <height-Mahler measure formula> is
$$
H(\alpha)^e=a_e\prod_{j=1}^e\max(1,|\alpha_j|).
$$
Its finite-place part follows from multiplicativity of the nonarchimedean polynomial coefficient norm: a primitive integer polynomial has norm one at each prime, so factoring it bounds the product of the roots' local maxima by the inverse leading-coefficient norm. The archimedean part is the displayed product over conjugates. Combining them with the product formula gives the identity.
Expanding the polynomial now bounds each integer coefficient by
$$
|a_k|\leq\binom ek a_e\prod_j\max(1,|\alpha_j|)\leq2^D B^D.
$$
There are finitely many degrees and finitely many such integer coefficient lists, each with finitely many roots. Hence there are only finitely many possible $\alpha$. This proves the <minimal-polynomial coefficient proof of Northcott finiteness> for algebraic numbers.
On the projective chart $a_j\ne0$, each ratio $\alpha_i=a_i/a_j$ has degree at most the point's field degree and
$$
H(\alpha_i)\leq\prod_v\left(\frac{\max_k|a_k|_v}{|a_j|_v}\right)^{n_v/d}=H(P).
$$
The equality uses the product formula. Each coordinate ratio therefore belongs to a finite set, and there are finitely many charts. This proves the projective <Northcott theorem>. Restricting to $V$ gives the result for a chosen embedding. For an ample line-bundle height, the comparison $mh_L=h_i+O(1)$ reduces its bounded-height sets to this case.
Both qualifications matter: a degree bound cannot be dropped, since infinitely many <roots of unity> have height one; and the finiteness assertion need not hold for a nonample line bundle, such as the trivial bundle's bounded height on a variety with infinitely many rational points.
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