Understanding the definition of the order of an entire function

Understanding the definition of the order of an entire function

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 $\varepsilon > 0$
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 $\varepsilon>0$,
there exists some $\rho_0>0$, 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.