Part BCSIR NET June 2024convergent-is-not-constant

Convergent is not constant

Let $(a_{n})_{n} \geq 1$ be a bounded sequence in $R$ . Which of the following statements is FALSE?

A.if $l imin f_{n} \to \infty a_{n} = l im s u p_{n} \to \infty a_{n}$ , then $(a_{n})$ is convergent
B.if inf{ $a_{n}$ | n ≥ 1} $= l im s u p_{n} \to \infty a_{n}$ , then $(a_{n})$ is convergent
C.if sup{ $a_{n}$ | n ≥ 1} $= l imin f_{n} \to \infty a_{n}$ , then $(a_{n})$ is constant✓
D.if sup{ $a_{n}$ | n ≥ 1} = inf{ $a_{n}$ | n ≥ 1}, then $(a_{n})$ is constant

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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