Rich Uncle Skeleton
Member
The "for all epsilon" in the definition of the limit is not a juvenile trick. It's the whole point! Maybe my choice of notation is confusing. If you prefer, say A = B + d, and then apply the definition of the limit with, in the traditional notation, epsilon = d/3 > 0.Ave22 said:This one feels like juvenile trickery showing that epsilon < 2epsilon/3 since we can pick epsilon to be whatever we want. But I suppose it works out, because it doesn't work when A > B, and it does when A = B and A < B.
If the proof doesn't feel right to you, draw a number line and mark A and B with A to the right of B. Then certainly you can draw tiny, disjoint open intervals around those two points. For large enough n, all the A_n will be trapped in the interval around A and all the B_n will be trapped in the one around B (and thus, to the left of the A_n). Hence the contradiction.
Now go back and formalize this: Give a name to the distance between A and B. Decide how tiny those intervals should be. Refer to the rigorous definition of the limit and apply it to show that the terms must lie inside the intervals. If you go through it all carefully, you'll see that what I wrote before is simply a formal version of the picture on the number line.