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. 

Hint:
Suppose this holds for $n=2$ (prove this base case yourself). Then 
$$\frac{(a_1+...a_k)+a_{k+1}}{(b_1+...+b_k)+b_{k+1}}\le \max\left(\frac{a_1+...+a_k}{b_1+...+b_{k}}, \frac{a_{k+1}}{b_{k+1}}\right)$$
