Vector space of infinite sequences in $\Bbb R$ [closed]

linear-algebra elementary-set-theory

How do you show that the set of infinite sequences with entries in $\Bbb R$ is uncountably infinite-dimensional as a vector space over $\Bbb R$.


Step 1: There is a family $\{A_r\mid r\in\Bbb R\}$ such that $A_r\subseteq\Bbb N$, and for $r\neq r'$ the intersection $A_r\cap A_{r'}$ is finite.

Step 2: Consider the characteristic functions of $A_r$ from the family of Step 1. Can they be linearly dependent?

(Another option is to use a diagonal argument to show that given any countable family of vectors, there is one not spanned by them. The above method has the benefit of showing that the dimension is as large as it can be, not just uncountable.)

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