Let $f: \mathbb C \to \mathbb C$ be an entire function. The order of $f$ is defined by $$\lambda=\limsup_{r \to \infty} \frac{\log \log M(r)}{\log r}, $$ where $$M(r)=\max_{|z|=r} |f(z)| .$$

The author claims that


  "According to this definition $\lambda$ is the smallest number such that
      $$M(r)\leq e^{r^{\lambda+\varepsilon}} $$ for any given $\varepsilon &gt; 0$ as soon as $r$ is sufficiently large."


Why is this true?

My attempt:

We know that $$\lambda=\lim_{\rho \to \infty} \sup_{r \geq \rho} \frac{\log \log M(r)}{\log r}. $$

From the definition of the limit we have that for any $\varepsilon&gt;0$, there exists some $\rho_0&gt;0$, such that $$\left\lvert \sup_{r \geq \rho} \frac{\log \log M(r)}{\log r}-\lambda \right\rvert \leq \varepsilon ,$$ for every $\rho \geq \rho_0$. In other words $$\frac{\log \log M(r)}{\log r} \leq \lambda+\varepsilon $$ for every $r \geq \rho_0$. From here it is easy to see that $$M(r)\leq e^{r^{\lambda+\varepsilon}}, $$ for all $r \geq \rho_0$. I cannot see why $\lambda$ is the smallest number with this property.

Thanks in advance.


Question: Are comments considered a form of documentation?
Answer: They are a form of documentation, but remember that documentation is in the eye of the beholder....


For some, self documenting code is enough.  But that assumes a level of technical detail as the customer.  We should be careful thinking that this is enough, because our ego may tell us "It is obvious what this code is doing" but time can prove otherwise.  It also assumes you know in advance the skills of the reader.
For those looking at source code but with less technical expertise, comments could be ok.  But that assumes someone is looking at the source code.
If you're technical, but lacking the time to read all the source code, a technical manual could be what's required.
An if the user lacks technical skills, but just needs to know what is happening, user documentation is what's needed.


So the real question is who is your customer?  If you are, then self documenting code or comments is enough.  If it's for someone else, you might want to broaden how you document.