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Given a certain set $E$ and a sequence $a_n$ whose limit is in $E$, we have to show that $E$ is closed.<\/p>\n
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 $a\\in E$.<\/p>\n
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.
\nSomehow 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.<\/p>\n
A:<\/p>\n
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.<\/p>\n
Sodium balance during acute and chronic oral amiloride infusions in humans.
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