According to axiom 4 in the textbook, local non satiation doesn't rule out the possibility that the preferred choice may involve less of all commodities. In my class my professor gave a little different definition of it, and after thinking a bit of it, I have the following question.

Let's assume the individual actually prefer less of all the commodities than those in bundle, say, A. Additionally, let's suppose that there exists an epsilon greater than zero such that the consumer reaches the bundle B, which has zero of each good. That's, the consumer prefers the bundle B=(0,0) (if we just assume the 2 good case) to A. According to my initial assumption and Axiom 4, there should be another bundle which is preferred to B=(0,0). But since negative amount of goods are not allowed, then there is no other bundle preferred to B=(0,0) which violates Axiom 4. Then, what I think is that Local Non satiation does rule out bundles which contain less of all commodities because, again, the bundle B=(0,0,...,0) always belongs to the consumption set.

Is my reasoning correct???