Why is the real number line not compact under the nested closed sets definition?

Conditions

• The ambient space is the real number line ℝ.
• Compactness is defined via nested closed nonempty subsets having a common point.
• Eₙ = [n, $\infty$) for $n = 1$, 2, 3, ...

Reasoning, step by step

1. Define Eₙ = [n, $\infty$) for positive integers n.
2. Verify that each Eₙ is a closed, nonempty subset of ℝ.
3. Observe that E₁ ⊃ E₂ ⊃ E₃ ⊃ forms a nested sequence.
4. Calculate the intersection of all Eₙ, which is empty because no real number is greater than or equal to every positive integer.
5. Conclude that ℝ fails the compactness definition.

Example

The video shows the intervals [1,$\infty$), [2,$\infty$), and [3,$\infty$) highlighted sequentially on a number line, then writes ⋂_{$n=1$}^{$\infty$} Eₙ = ∅ to demonstrate the empty intersection.

Common misconceptions

• Thinking that unbounded sets can still be compact if they are closed.
• Assuming that the intersection of nested closed sets is always nonempty without checking if the sets are bounded.

Watch the explanation

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