MthT 430 Projects Chap 8f - limsup and liminf

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(See also Spivak Chapter 8 - Problem 18) Let {x_k} be a bounded sequence. We define the limit superior (limsup) and limit inferior (liminf) of the sequence to be

limsup k → ∞  xk = lim k→∞  ⎛ ⎝ sup n ≥ k  xn ⎞ ⎠ , liminf k → ∞  xk = lim k→∞  ⎛ ⎝ inf n ≥ k  xn ⎞ ⎠ .

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(P13-BISHL) shows that both limsup_{k → ∞}x_k and liminf_{k → ∞} x_k exist.

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Show that

limsup k → ∞  xk = A

if and only if for every ϵ > 0,

⎧ ⎪ ⎨ ⎪ ⎩ xk > A + ϵ for at most finitely many k, xk > A − ϵ for infinitely many k.

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Show that if {x_k} and {y_k} are bounded sequences, then

limsup k → ∞  (xk + yk) ≤ limsup k → ∞  xk + limsup k → ∞  yk.

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Give an example of bounded sequences {x_k} and {y_k} such that

limsup k → ∞  (xk + yk) < limsup k → ∞  xk + limsup k → ∞  yk.

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Show that if {x_k} and {y_k} are bounded sequences, then

liminf k → ∞  xk + limsup k → ∞  yk ≤ limsup k → ∞  (xk + yk).

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The general result is that for two bounded sequences {x_k} and {y_k},

liminf k → ∞  xk + liminf k → ∞  yk ≤ liminf k → ∞  (xk + yk) ≤ liminf k → ∞  xk + limsup k → ∞  yk ≤ limsup k → ∞  (xk + yk) ≤ limsup k → ∞  xk + limsup k → ∞  yk.

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On 22 Aug 2014, 01:37.