A sequence of closed bounded (not necessarily nested) intervals $I_1, I_2, I_3,...$ with the property that

Question

Abbot's Understanding Analysis Problem 1.4.8 (d)

If the following statement is possible, give an example; if it is not, provide a compelling argument as to why it is not possible.

A sequence of closed bounded (not necessarily nested) intervals $I_1, I_2, I_3,...$ with the property that

(1) $$\cap_{n=1}^{N} I_n \neq \emptyset, \quad \forall N \in \mathbb{N},$$

and

(2)$$\cup_{n=1}^{\infty} I_n = \emptyset.$$

It seems to me that the statement is not possible. I considered the sequence of nested intervals $(0,1/n]$ for all $n \in \mathbb{N}.$ I know that this does not satisfy the given conditions (it is not closed), but I went through it simply as an exercise. This sequence of open intervals would satisfy both conditions. Since the sequence of intervals must all include zero (as each interval must be closed), it would not otherwise satisfy the properties. Does the falsity of the statement have anything to do with the properties of open and closed intervals?

I tried considering sequences of intervals that were not nested, yet was unable to produce such a sequence that would satisfy both properties.

If the statement is not true, would you give an complete explanation as to why it is impossible? It would seem to me (if it is not true) that it has something to do with the fact that the intervals are closed.

Moreover, it seems to me that a sequence that fulfills both properties cannot be nested, as for all nested, closed sequences of intervals, $$\cup_{n=1}^{\infty} I_n \neq \emptyset.$$

Answer 1

It is not possible. Assume $\bigcap_{n=1}^N I_n \ne \emptyset, \forall N \in\mathbb{N}$ but $\bigcap_{n=1}^\infty I_n = \emptyset$.

Notice that $\bigcap_{n=1}^\infty I_n \subseteq I_1$ so we have

$$I_1 = I_1 \setminus \emptyset = I_1 \setminus \left(\bigcap_{n=1}^\infty I_n\right) = \bigcup_{n=1}^\infty (I_1 \setminus I_n)$$

The sets $I_1 \setminus I_n$ are open in $I_1$ so $(I_1 \setminus I_n)_{n=1}^\infty$ is an open cover of the compact set $I_1$.

Therefore, there exists $N \in \mathbb{N}$ such that $I_1 = \bigcup_{n=1}^N(I_1 \setminus I_n)$. Again taking the complement in $I_1$ gives $\bigcap_{n=1}^N I_n = \emptyset$ which is a contradiction with the assumption.

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Note that it is possible if $I_n$ are not assumed to be closed:

$$I_n = \left\langle0, \frac1n\right]$$

or bounded: $$I_n = \left[n, +\infty\right\rangle$$

Answer 2

Hint (assuming the exercise is motivated by a version of Cantor's intersection Theorem): If $(2) \cap_{n=1}^\infty I_n=\emptyset$, then consider $A_N:=\cap_{n=1}^NI_n, $ for $ N\in \mathbb N$, which are non empty nested, closed and bounded sets, and $\cap_{N=1}^M A_N=\cap_{n=1}^MI_n$ for all $M$.

Answer 3

Here is a proof that such a sequence is impossible, without assuming knowledge of open covers and compact sets.

Let $(I_n)_{n=1}^\infty$ be a sequence of closed, bounded intervals (not necessarily nested) such that $\bigcap_{n=1}^NI_n\not=\emptyset$ for all $N\in\mathbb N$. We show that $\bigcap_{n=1}^\infty I_n\not=\emptyset$.

Proof: Define the following sequence:

$$\left(\bigcap_{n=1}^NI_n\right)_{N=1}^\infty=\left(I_1,I_1\cap I_2,I_1\cap I_2\cap I_3,...\right)\text.$$

Note that these intervals are nested, since $A\cap B\subset A$ for any sets $A$ and $B$. By the Nested Interval Property, we have the following:

$$\bigcap_{n=1}^\infty I_n=\bigcap_{N=1}^\infty\left(\bigcap_{n=1}^NI_n\right)\not=\emptyset\text,$$

where the first equality is true because

$$I_1\cap I_2\cap I_3\cap...=I_1\cap(I_1\cap I_2)\cap(I_1\cap I_2\cap I_3)\cap...\text.$$

Answer 4

Let $I_j:=[a_j,b_j]$ for $j=1,2,\dots$.
Suppose that $$\bigcap_{n=1}^{N}I_j\neq\emptyset$$ for all $N\in\mathbb{N}.$
Let $c_N:=\max\{a_1,\dots,a_N\}$ and $d_N:=\min\{b_1,\dots,b_N\}.$
Then, $$\bigcap_{n=1}^{N}I_j=[c_N,d_N].$$ Since $$[c_1,d_1]\supset [c_2,d_2]\supset [c_3,d_3]\supset\dots,$$ $$\bigcap_{n=1}^{\infty} [c_n,d_n]\neq\emptyset$$ by Theorem 1.4.1 (Nested Interval Property) on p.20.
So, $$\bigcap_{n=1}^{\infty}I_j=\bigcap_{N=1}^{\infty}\bigcap_{n=1}^{N}I_j=\bigcap_{N=1}^{\infty}[c_N,d_N]\neq\emptyset.$$