Consider the distinct embeddings $\sigma_\ell\colon K \to \mathbb{C}$ of the number field $K$ in the complex numbers (up to equality on $K$, not just up to their image): by standard facts in Galois theory, there are exactly $h$ (i.e. $[K:\mathbb{Q}] =: h$) of them, and they are linearly independent over $\mathbb{C}$ (see: linear independence of characters). Now each $\sigma_\ell$ is a $\mathbb{Q}$-linear map $a_1\beta_1 + \cdots + a_h\beta_h \mapsto b_{\ell,1}\, a_1 + \cdots + b_{\ell,h}\, a_h$ for some complex coefficients $b_{\ell,j} := \sigma_\ell(\beta_j)$: linear independence tells us that the $h\times h$ complex matrix $(b_{\ell,j})$ is invertible, say $\sum_{\ell=1}^h c_{i,\ell}\, b_{\ell,j} = \delta_{i,j}$, and then $a_i = \sum_{\ell,j} c_{i,\ell}\, b_{\ell,j}\, a_j$ for any complex $a_1,\ldots,a_h$ and in particular for rational ones, meaning $a_i = \sum_\ell c_{i,\ell} \, \sigma_\ell(\alpha)$ for $\alpha\in K$, which gives us $|a_i| \leq c\cdot \max|\sigma_\ell(\alpha)|$ where $c = \lceil\sum_\ell |c_{i,\ell}|\rceil$, as claimed.
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