So define the set of finite sequences to be $S={a_1,a_2,\cdots}$ where $a_k$ are in real numbers and only finitely many of them are non-zero. The set of infinite sequences is defined similarly except that we can have infinitely many non-zero terms. How do I prove that there does not exist a bijection between these two sets?
Bijection between finite and infinite sequences over Reals.
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real-analysis
sequences-and-series
set-theory
cardinals
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0As you tagged this as *linear algebra*, could it be that instead of bijection you mean isomorphism of real vector spaces? – 2017-02-06
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0I was wondering how I show that the set of infinite sequences with entries in R is uncountably infinite- dimensional as a vector space over R. – 2017-02-06
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0@JohnvonNeumann Isn't the set of infinite sequences of real numbers manifestly a *countably* infinite-dimensional vector space over R? – 2017-02-06
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0spaceisdarkgreen : not so manifestly though – 2017-02-06
1 Answers
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You don't, because there exist such bijections. The first one is (essentially) $\displaystyle\bigcup_{n\in \mathbb{N}} \mathbb{R}^n$, which has the same cardinality as $\mathbb{R}$, and the second one is $\mathbb{R}^\mathbb{N}$, which also has the same cardinality as $\mathbb{R}$.