## What does “$E$ is not bounded above” mean? I am confused. “Principles of Mathematical Analysis” by Walter Rudin Theorem 3.17.

I am reading Walter Rudin’s “Principles of Mathematical Analysis”.

There are the following definition and theorem and its proof in this book.

Definition 3.16:

Let $$\{ s_n \}$$ be a sequence of real numbers. Let $$E$$ be the set of numbers $$x$$ (in the extended real number system) such that $$s_{n_k} \rightarrow x$$ for some subsequence $$\{s_{n_k}\}$$. This set $$E$$ contains all subsequential limits, plus possibly the numbers $$+\infty$$, $$-\infty$$.

Put $$s^* = \sup E,$$ $$s_* = \inf E.$$

Theorem 3.17:

Let $$\{s_n \}$$ be a sequence of real numbers. Let $$E$$ and $$s^*$$ have the same meaning as in Definition 3.16. Then $$s^*$$ has the following two properties:

(a) $$s^* \in E$$.

(b) If $$x> s^*$$, there is an integer $$N$$ such that $$n \geq N$$ implies $$s_n < x$$.

Moreover, $$s^*$$ is the only number with the properties (a) and (b).

Of course, an analogous result is true for $$s_*$$.

Proof:

(a)
if $$s^* = +\infty$$, then $$E$$ is not bounded above; hence $$\{s_n\}$$ is not bounded above, and there is a subsequence $$\{s_{n_k}\}$$ such that $$s_{n_k} \to +\infty$$.

If $$s^*$$ is real, then $$E$$ is bounded above, and at least one subsequential limit exists, so that (a) follows from Theorems 3.7 and 2.28.

If $$s^* = -\infty$$, then $$E$$ contains only one element, namely $$-\infty$$, and there is no subsequential limit. Hence, for any real $$M$$, $$s_n > M$$ for at most a finite number of values of $$n$$, so that $$s_n \to -\infty$$.

This establishes (a) in all cases.

I cannot understand the following argument:

(a)
if $$s^* = +\infty$$, then $$E$$ is not bounded above; hence $$\{s_n\}$$ is not bounded above, and there is a subsequence $$\{s_{n_k}\}$$ such that $$s_{n_k} \to +\infty$$.

What does “$$E$$ is not bounded above” mean?
p.12, Rudin wrote “It is then clear that $$+\infty$$ is an upper bound of every subset of the extended real number system”.
And $$E$$ is a subset of the extended real number system.

## Can I get a driver for my BT v4 adapter to the Android 8.0 and above?

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My questions are:

Can I get a driver for my BT v4 adaptor to the Android 8.0 and above?

Or should I buy a different adaptor which is compatible with the Android ?

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## Prove that the sequence is bounded above.

I have proved that the induction hypothesis is true for n=1. However when trying to prove that the hypothesis is true for n=k+1 when we assume that it is true for n=k, I am getting stuck. I have tried various manipulations to prove that the hypothesis is true for n=k+1. Kindly help.

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Death’s Embrace (Ex): At 8th level, you heal damage instead of taking damage from channeled negative energy. If the channeled negative energy targets undead, you heal hit points just like undead in the area.

So does that mean a cleric with the death domain is healed by both positive AND negative energy channeling, or does this in effect make the cleric react to positive energy like an undead and take damage? Further, is that cleric affected by healing/harm spells in the same manner?

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If it matters, I want to do this in python.

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Is there any way to get Drive to actually sort things alphabetically as would normally be expected in any standard filesystem?

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The scenario is I have to submit the data to the 1st API if there is no error response found then submit the data to 2nd API so now I have created a submit handler and attached the handler to the webform. Now the issue is when I got an error response message while submitting the data to 1st API and showing the message using drupal_set_message() function everything is working fine but the form validation js not loading at this point.

The used code is like the below:

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