Understanding a detail in the proof of bin packing approximation
We have $u = 1 - e$ where $u$ is used space and $e$ is empty space.

Since for at least $FF - 1$ bins we have $e &lt; \delta$ (which you understand why), for those we also have $1- e &gt; 1 - \delta$ or $u &gt; 1-\delta$.

So in at least $FF - 1$ bins we use at least $1 - \delta$ space. Since each bin has maximum size $1$ we can also say that at least $FF - 1$ bins are at least $1 - \delta$ fraction full. In other words, we have at least $(FF - 1)(1 - \delta)$ bins worth of 'stuff', so that must be less than or equal to the optimal amount of bins we need to use (or else we'd have more stuff than that could fit).