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Decide if the set of the convergent sequences is a subspaces of the vector space of the infinite sequences $(a_i)^{\infty}_0$.

My solution:

For the convergence we have to satisfy following condition: $\lim_{n\to \infty} a_n = A$, so every sequence that belongs to the set of the convergent sequences must have a finite limit.

1) $\lim_{n\to \infty} (a_n + b_n )= \lim_{n\to \infty} a_n + \lim_{n\to \infty} b_n = A + B$ and that is a finite limit.

2) For $r\in \mathbb R$, $\lim_{n\to \infty} (ra_n) = rA$, the limit exists, so it is ok.

Sorry for my english.

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    $\begingroup$ If $r\in \mathbb R$, then $r$ is not infinite. And if the limit is $rA$, then the limit exists, so the sequence is convergent. $\endgroup$ – Jonas Meyer Feb 6 '17 at 14:41
  • $\begingroup$ Real numbers are an infinite set, uncountable set, isn't it right? So why $r$ can not be infinite? $\endgroup$ – Leif Feb 6 '17 at 14:53
  • $\begingroup$ There are uncountably many real numbers. All of them are finite. Infinity is not a number. Except for that confusion, your answer is correct. $\endgroup$ – Ethan Bolker Feb 6 '17 at 15:00
  • $\begingroup$ @Astrid There is no "is finite" or "is infinite" for a real number. The real numbers are all expressible as decimal expansions, and none of them could be considered "infinite." You can talk about the extended real numbers which adjoin elements that act like inifinity, and then talk about convergence to the adjoined elements, however. $\endgroup$ – rschwieb Feb 6 '17 at 15:01
  • $\begingroup$ Ok, so infinite can be a set of numbers but not a number itself. But I am little confused now, is it a subspace or not? If $r$ is finite, then it is convergent. $\endgroup$ – Leif Feb 6 '17 at 15:07

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