On proving the convergence of $1/n^2\sum_{1\le k\le n}\varphi(k)$ Let $$\Phi_n=\frac{1}{n^2}\sum_{k=1}^n\varphi(k).$$ How one can show that  $\Phi_n$ is convergent sequence? 
(Here, $\varphi$ denotes the  Euler's  totient function.)
And please, without any monster asymptotic estimations.
Thank you!
EDIT:
Can we actually prove the convergence without finding the limit?  

We certainly have $\varphi(k)\le k \space\forall k\ge 1$, and $\lim_{n\to \infty}\frac{1}{n^2}\sum_{k=1}^nk=\frac{1}{2}$. This shows that $\lim \sup \Phi_n\le 1/2$. It is known that $\Phi_n\sim \frac{3}{\pi^2}$.
