Part CCSIR NET June 2024uniform-on-every-compact-piece-is-not-uniform-overall

Uniform on every compact piece is not uniform overall

Let $f : [0, 1) \to [1, \infty)$ be defined by f(x) = 1/(1 − x). For n ≥ 1, let $p_{n} (x) = 1 + x + \dots + x^{n}$ . Then which of the following statements are true?

A.f(x) is not uniformly continuous on [0, 1)✓
B.The sequence $(p_{n} (x))$ converges to f(x) pointwise on [0, 1)✓
C.The sequence $(p_{n} (x))$ converges to f(x) uniformly on [0, 1)
D.The sequence $(p_{n} (x))$ converges to f(x) uniformly on [0, c] for every 0 < c < 1✓

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′

Uniform on every compact piece is not uniform overall

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