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Go to best e-learning portal siteQ:
Given a certain set $E$ and a sequence $a_n$ whose limit is in $E$, we have to show that $E$ is closed.
Consider the set $E$ of all sequences $(a_n)_{n\in\Bbb N}$ satisfying $a_n>0$, for all $n\in\Bbb N$, and $$\lim_{n\to\infty} a_n=0.$$ Let $a=(a_n)_{n\in\Bbb N}$ be a sequence in $E$, and let $(b_n)_{n\in\Bbb N}$ be a sequence of positive reals whose limit is $a$. We have to show that $a\in E$.
I took a look at the textbook I have (guess which one it is). This exercise is just a step in the proof of a result, Theorem 1.2.7. However, the argument given in this textbook is completely different from mine.
Somehow I have to proof that the real $0$ is an accumulation point of $E$, and that $\forall\epsilon\gt0\exists n\in\Bbb N:d(b_n,0)\lt\epsilon$, but I don’t know how to tackle this.
A:
For a sequence in $E$, its limit is in $E$. However, $0$ is not a limit of $b_n$, so $0$ is not a limit of the sequence.
Sodium balance during acute and chronic oral amiloride infusions in humans.
Oral amiloride (AM) produces sodium (Na) conservation with little or no effect on potassium (K) and magnesium (Mg) balance. The present studies examined the
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