Paper 1, Section I, $3 F$

(a) State, without proof, the Bolzano-Weierstrass Theorem.

(b) Give an example of a sequence that does not have a convergent subsequence.

(c) Give an example of an unbounded sequence having a convergent subsequence.

(d) Let $a_{n}=1+(-1)^{\lfloor n / 2\rfloor}(1+1 / n)$, where $\lfloor x\rfloor$ denotes the integer part of $x$. Find all values $c$ such that the sequence $\left\{a_{n}\right\}$ has a subsequence converging to $c$. For each such value, provide a subsequence converging to it.

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