The lazy caterer's problem is to figure out the maximum number of pieces formed by $n$ straight cuts of a pizza. Any time two cuts meet new pieces are generated, so for maximum number of pieces it makes sense that all possible meetings are present. My question is how can this be guaranteed to happen. The wikipedia article says that
A cut line can always cross over all previous cut lines, as rotating the knife at a small angle around a point that is not an existing intersection will, if the angle is small enough, intersect all the previous lines including the last one added.
What would be a mathematical proof of the above fact?
Edit: One of the earlier answers which was deleted was as follows: Given $n-1$ lines in the plane all intersecting each other separately it is always possible to draw an $n$th line which too separately intersects the others by selecting an appropriately different slope for it then of the others. A circle may then be constructed containing all the intersections which can then be scaled to the size of desired pizza.
Was this argument correct? It seems so to me, and I am confused as to why it was deleted.