Part CCSIR NET December 2024shifting-is-tame-and-rescaling-is-not

Shifting is tame and rescaling is not

Let $f : R \to R$ be a uniformly continuous function. For each positive integer n and $x \in R$ , let $g_{n}$ and $h_{n}$ be defined by $g_{n} (x) = f (x + 1/ n)$ and $h_{n} (x) = f ($ nx). Which of the following statements are necessarily true?

A. $(g_{n})_{n} \geq_{1}$ converges uniformly on any compact subset of $R$ but not on $R$ .
B. $(g_{n})_{n} \geq_{1}$ converges uniformly on $R$ .✓
C.There exists a subsequence of $(h_{n})_{n} \geq_{1}$ that converges uniformly on $R$ .
D.For every compact subset $K \subseteq R$ , there exists a subsequence of $(h_{n})_{n} \geq_{1}$ that converges uniformly on K.

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.

See pricing

50 are analysed free — try those first.

Related counterexample: fₙ → f uniformly ⇒ fₙ′ → f′

Shifting is tame and rescaling is not

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