How this proof of fractional knapsack works?
The following condition is implicitly included in the question.

$$\dfrac{v_1}{w_1}, \dfrac{v_2}{w_2}, \cdots, \dfrac{v_n}{w_n} \text{ is in the descending order.}$$

Let $W$ be the total weight to be filled. The greedy algorithm for fractional knapsack problem is the following procedure.

Let $k$ loop through $1, 2, ..., n$ in that order.


Set $$x_k = \dfrac{W- \sum_{1\le i\lt k} x_i}{w_i}$$
If $x_k&lt;1$, set $x_l=0$ for all $l&gt;k$. Break the loop.




$$\dfrac{v_i}{w_i}(x_i-y_i)\ge\dfrac{v_j}{w_j}(x_i-y_i),\tag{????}$$

There are three cases for $i$.


The case when i &lt; j, i.e., $\dfrac{v_i}{w_i}\ge\dfrac{v_j}{w_j}$. Since $j$ is the first index s.t. $x_j&lt;1$, $x_i=1$, i.e., $x_i-y_i\ge0$. 
The case when i = j. Both sides are 0.
The case when i > j, i.e., $\dfrac{v_i}{w_i}\le\dfrac{v_j}{w_j}$. Since $j$ is the first index s.t. $x_j&lt;1$, according to the definition of the greedy algorithm, $x_i=0$. That means $x_i-y_i\le0$.