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

Cauchy is preserved because the line is complete

Let $f : R \to R$ be a non-constant continuous function. Which of the following statements is necessarily true?

A.For every bounded subset $A \subseteq R, f$ ⁻ $^{1} (A)$ is a bounded subset of $R$ .
B.For every Cauchy sequence $(x_{n})_{n} \geq_{1}$ in $R, (f (x_{n}))_{n} \geq_{1}$ is a Cauchy sequence in $R$ .✓
C.There exists $x \in R$ such that f(x) = x.
D.There exists $x \in R$ 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.

50 are analysed free — try those first.

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