In Russia there is a kind of doll called a matrushka doll. You pick it up and notice it comes apart in the center. Separating the two halves you find an identical doll inside. Then you notice this inside doll also comes apart in the center. Separating the two halves, you find yet another identical doll inside. This goes on quite a while until the final doll is in one piece. The nested interval lemma is like a matrushka doll except the process never stops. It involves a sequence of intervals, the first containing the second, the second containing the third, the third containing the fourth and so on. The fundamental question is whether there exists a point in all the intervals. Sometimes there is such a point and this comes from completeness.
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Then there exists a point, c ∈ ℝ which is an element of every Ik. If the diameters (length) of these intervals, denoted as diam
Proof:Since Ik ⊇ Ik+1, this implies
| (3.4) |
Consequently, if k ≤ l,
| (3.5) |
Now define
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By the first inequality in 3.4, and 3.5
| (3.6) |
for each k = 1,2
For the last claim, suppose there are two points x,y in the intersection. Then
This is really quite a remarkable result and may not seem so obvious. Consider the intervals Ik ≡