proof of correctness for greedy knapsack algorithm
Slides are not a substitute for a textbook or careful exposition.  This is especially true when reading a proof.  Slides are intended for presentation in real-time, and as a result often leave out some details.  If there is some aspect you don't understand when reading a set of slides, usually the best thing to do is to find a proper written exposition of the subject -- usually a textbook (but sometimes written lecture notes can be adequate, if written in enough detail).

In this case, on the previous slide, the statement of the theorem mentions "nonincreasing order of $p_i/w_i$", which means that $p_1/w_1 \ge p_2/w_2 \ge \cdots$.  In particular, $p_k/w_k \ge p_i/w_i$ when $k&lt;i$.  The inequality you are referencing then follows.