Ratio Inequality How can I prove that,
$$\frac{a_{1}+a_{2}+\dots+a_{n}}{b_{1}+b_{2}+\dots+b_{n}} \le \max_i\left\{\frac{a_{i}}{b_{i}}\right\}$$ 
where $1 \le i \le n$, and $a_{i} \neq a_{j}$ and $b_{i} \neq b_{j}, \forall i \neq j$  
Edit
I have figured out that the above assumptions about $a_{i}$, and $b_{i}$ are not needed. 

The $a_i$ can be arbitrary real numbers, but the $b_i$ need to
be positive. Then
$$ \tag{*}
  a_j \le b_j \cdot \left\{ \max_i \frac{a_i}{b_i} \right \}
  \quad \text{for $j =1, \ldots, n$}
$$
and adding these gives the desired inequality.
If the $b_i$ are not required to be positive then the
inequality must not hold, a 
counter-example is
$$
   \frac {2 - 1}{3 - 2} > \max \left\{ \frac{2}{3}, \frac{-1}{-2}  \right \} \, .
$$
