- #1
christianrhiley
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Definitions: Let {x[n]} be a bounded sequence in Reals.
We define {y[k]} and {z[k]} by
y[k]=sup{x[n]: n \geq k}, z[k]=inf{x[n]: n \geq 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 \geq k} and S[k+1] = {x[n]: n \geq k + 1}
S[k] is a \subset S[k+1], and if sup(S[k]) \leq sup(S[k+1]), it follows that S[k] \leq S[k+1]. We conclude that {y[k]} is decreasing.
(iii)similar to part (ii) except inf(S[k+1]) \leq inf(S[k])
Note: inf(A) \leq inf(B) and sup(B) \leq 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.
We define {y[k]} and {z[k]} by
y[k]=sup{x[n]: n \geq k}, z[k]=inf{x[n]: n \geq 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 \geq k} and S[k+1] = {x[n]: n \geq k + 1}
S[k] is a \subset S[k+1], and if sup(S[k]) \leq sup(S[k+1]), it follows that S[k] \leq S[k+1]. We conclude that {y[k]} is decreasing.
(iii)similar to part (ii) except inf(S[k+1]) \leq inf(S[k])
Note: inf(A) \leq inf(B) and sup(B) \leq 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.
