Prove that $\mathbb{R}^\infty$ is an infinite-dimensional vector space

Question

I am busy studying for a Linear Algebra test and came across this question in Section 4.4 of Elementary Linear Algebra (Application Version) $11^{th}$ edition.

This is not part of my work that needs to be covered in the test, but I thought it would be interesting to see how this can be proven. Can anyone please show me how this can be done? :)

Answer 1

Tne vectors $e_n $ where $$e_n (k) =\begin{cases} 1 \mbox{ if } n=k \\ 0 \mbox{ if } n\neq k\end{cases} $$ are linearly independent.

Answer 2

$$A = \{(1,0,0,...), (0,1,0,0,...),...\}$$ span $\Bbb R^\infty$ and it is very easy to check they are all linearly independent.

Added: $A$ is an infinite set, so $dim(\Bbb R^\infty)$ is infinite by definition of $dim()$.