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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 \(\displaystyle B\) is an upper bound for every \(\displaystyle n\)-tail of \(\displaystyle \{ x_n \}\), so \(\displaystyle \overline{ x_n } = \text{sup} \{ x_k \ : \ k \geq n \} \leq B\). Thus \(\displaystyle \lim_{ n \to \infty } \overline{ x_n } \leq B\) ... ... "

My question is as follows:

Can someone please explain exactly how it follows that \(\displaystyle \lim_{ n \to \infty } \overline{ x_n } \leq B\) ... that is, how it follows that \(\displaystyle \overline{ \lim_{ n \to \infty } } x_n \leq B\) ...

(... ... 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 9248

Hope 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 \(\displaystyle B\) is an upper bound for every \(\displaystyle n\)-tail of \(\displaystyle \{ x_n \}\), so \(\displaystyle \overline{ x_n } = \text{sup} \{ x_k \ : \ k \geq n \} \leq B\). Thus \(\displaystyle \lim_{ n \to \infty } \overline{ x_n } \leq B\) ... ... "

My question is as follows:

Can someone please explain exactly how it follows that \(\displaystyle \lim_{ n \to \infty } \overline{ x_n } \leq B\) ... that is, how it follows that \(\displaystyle \overline{ \lim_{ n \to \infty } } x_n \leq B\) ...

(... ... 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 9248

Hope that helps ...

Peter

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