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I am reading Charles G. Denlinger's book: "Elements of Real Analysis".
I am focused on Chapter 2: Sequences ... ...
I need help with the proof of Theorem 2.9.6 (b)Theorem 2.9.6 reads as follows:View attachment 9245
View attachment 9246
In the above proof of part (b) we read the following:
" ... ... Then $$B$$ is an upper bound for every $$n$$-tail of $$\{ x_n \}$$, so $$\overline{ x_n } = \text{sup} \{ x_k \ : \ k \geq n \} \leq B$$. Thus $$\lim_{ n \to \infty } \overline{ x_n } \leq B$$ ... ... "My question is as follows:
Can someone please explain exactly how it follows that $$\lim_{ n \to \infty } \overline{ x_n } \leq B$$ ... that is, how it follows that $$\overline{ \lim_{ n \to \infty } } x_n \leq B$$ ...
(... ... apologies to steep if this is very similar to what has been discussed recently ... )
Hope someone can help ...
Peter
===============================================================================
It may help MHB readers to have access to Denlinger's definitions and notation regarding upper and lower limits ... so I am providing access to the same ... as follows:
View attachment 9247
View attachment 9248Hope that helps ...
Peter
I am focused on Chapter 2: Sequences ... ...
I need help with the proof of Theorem 2.9.6 (b)Theorem 2.9.6 reads as follows:View attachment 9245
View attachment 9246
In the above proof of part (b) we read the following:
" ... ... Then $$B$$ is an upper bound for every $$n$$-tail of $$\{ x_n \}$$, so $$\overline{ x_n } = \text{sup} \{ x_k \ : \ k \geq n \} \leq B$$. Thus $$\lim_{ n \to \infty } \overline{ x_n } \leq B$$ ... ... "My question is as follows:
Can someone please explain exactly how it follows that $$\lim_{ n \to \infty } \overline{ x_n } \leq B$$ ... that is, how it follows that $$\overline{ \lim_{ n \to \infty } } x_n \leq B$$ ...
(... ... apologies to steep if this is very similar to what has been discussed recently ... )
Hope someone can help ...
Peter
===============================================================================
It may help MHB readers to have access to Denlinger's definitions and notation regarding upper and lower limits ... so I am providing access to the same ... as follows:
View attachment 9247
View attachment 9248Hope that helps ...
Peter
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