Showing the equivalence of different definitions of the box-counting dimension

Question

I am trying to prove the statment "by taking logarithms...".

$\lim\limits_{\delta \rightarrow 0} \frac{log(N_{4\delta}(F))}{log(\frac{1}{4 \delta})} \leq \lim\limits_{\delta \rightarrow 0} \frac{log(N_{\delta}'(F))}{log(\frac{1}{4 \delta})}$

However I am not sure how to deal with the 4\delta and \delta in the usual definition of the box counting dimension, given below.

Answer 1

Since $$\lim_{\delta\to 0} \frac{\log (4\delta)}{\log \delta}=1$$ you can replace $\log 4\delta$ with $\log \delta$ in any denominator here. For example, $$\limsup_{\delta \to 0} \frac{\log(N_{4\delta}(F))}{\log(\frac{1}{4 \delta})} = \limsup_{\delta \to 0} \frac{\log(N_{4\delta}(F))}{\log(\frac{1}{ \delta})}$$