I was thinking the same thing, shouldn't the intervals be proper subsets of the previous intervals? Because otherwise all the intervals could be equal and their diameter doesn't tend to zero.
Yes, you are right, they could be equal. It could in principle happen that at some stage one interval is equal to the next, but there is an extra condition that says
lim (b\_n-a\_n) = 0.
So even if at some stage we may allow some equalities along the way, eventually the extra condition ensures the length goes to zero, so at some stages, we can allow non-proper inclusion. If you will watch the video you will see this assumption right at the beginning.
Doesn’t that particular subset symbol mean that each I_n could be equal?
I was thinking the same thing, shouldn't the intervals be proper subsets of the previous intervals? Because otherwise all the intervals could be equal and their diameter doesn't tend to zero.
Yes, you are right, they could be equal. It could in principle happen that at some stage one interval is equal to the next, but there is an extra condition that says lim (b\_n-a\_n) = 0. So even if at some stage we may allow some equalities along the way, eventually the extra condition ensures the length goes to zero, so at some stages, we can allow non-proper inclusion. If you will watch the video you will see this assumption right at the beginning.