I am reading the proof of Shannons Entropy in his paper of 1948. (http://math.harvard.edu/~ctm/home/text/others/shannon/entropy/entropy.pdf)
In the proof (Aprendix 2, page 29), I do not understand why
$|\frac{A(t)}{A(s)}-\frac{\log t}{\log s}|<2\epsilon$
implies that
$ A(t)=K \log t$
Where does the $K$ came from?
Since $\epsilon$ is arbitrarily small, Shannon has proved that
$\frac{A(t)}{A(s)} = \frac{\log(t)}{\log(s)} \quad \forall s,t \in \mathbb{Z}^+$
Re-arranging this identity gives
$\frac{A(t)}{\log(t)} = \frac{A(s)}{\log(s)} \quad \forall s,t \in \mathbb{Z}^+$
If we call this ratio $K$ then we have
$\frac{A(t)}{\log(t)} = K \quad \forall t \in \mathbb{Z}^+$
$\Rightarrow A(t) = K \log(t)\quad \forall t \in \mathbb{Z}^+$