(adsbygoogle = window.adsbygoogle || []).push({}); Definitions:Let {x[n]} be a bounded sequence in Reals.

We define {y[k]} and {z[k]} by

y[k]=sup{x[n]: n [tex]\geq[/tex] k}, z[k]=inf{x[n]: n [tex]\geq[/tex] k}

Claim:(i) Both y[k] and z[k] are bounded sequences

(ii){y[k]} is a decreasing sequence

(iii){z[k]} is an increasing sequence

Proof:(i) suppose y[k] and z[k] are not bounded. this implies x[n] is unbounded, a contradiction. therefore, we conclude that both y[k] and z[k] are bounded.

(ii)Let S[k] = {x[n]: n [tex]\geq[/tex] k} and S[k+1] = {x[n]: n [tex]\geq[/tex] k + 1}

S[k] is a [tex]\subset[/tex] S[k+1], and if sup(S[k]) [tex]\leq[/tex] sup(S[k+1]), it follows that S[k] [tex]\leq[/tex] S[k+1]. We conclude that {y[k]} is decreasing.

(iii)similar to part (ii) except inf(S[k+1]) [tex]\leq[/tex] inf(S[k])

Note:inf(A) [tex]\leq[/tex] inf(B) and sup(B) [tex]\leq[/tex] sup(A) have already been proven in an earlier exercise.

Where I Need Help:I need input regarding all three parts. I have made, at best, an informal sketch of a proof, and I would like some input on how to turn it into a rigorous proof.

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# Prove Sequence is Bounded

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