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Part BCSIR NET December 2024cauchy-is-preserved-because-the-line-is-complete

Cauchy is preserved because the line is complete

Let f:R→Rf : \mathbb{R} \to \mathbb{R} be a non-constant continuous function. Which of the following statements is necessarily true?

  1. A.For every bounded subset A⊆R,fA \subseteq \mathbb{R}, f⁻1(A)^{1}(A) is a bounded subset of R\mathbb{R}.
  2. B.For every Cauchy sequence (xn)n≥1(x_{n})_{n}\ge_{1} in R,(f(xn))n≥1\mathbb{R}, (f(x_{n}))_{n}\ge_{1} is a Cauchy sequence in R\mathbb{R}.✓ Correct
  3. C.There exists x∈Rx \in \mathbb{R} such that f(x) = x.
  4. D.There exists x∈Rx \in \mathbb{R} such that f(x) = 0.

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests hypothesis dropped.

See pricing

50 are analysed free — try those first.

The trap it tests

Hypothesis dropped

A theorem applied without checking one of its preconditions.

Drill statements like this

Related counterexample: Continuous on a bounded interval ⇒ bounded

More on this topic

The chapter behind this: Continuity vs uniform continuity vs Lipschitz — free to read

From Continuity and Differentiation › Continuity, uniform continuity, Lipschitz

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