In a book of metric space topology it is claimed that $\mathbb{Q}^{\mathbb{N}}\cap \ell^2$ is uncountable where $\ell^2$ is a space of square-summable real sequences. This means we have to show that the set all rational square-summable sequences is uncountable. I don't know to to start. Kindly help!
Proving $\mathbb{Q}^{\mathbb{N}}\cap \ell^2$ is uncountable.
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real-analysis
general-topology
metric-spaces
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3The intersection contains $S:=\prod\limits_{n=1}^\infty\,\Biggl(\left[-\frac{1}{n},+\frac{1}{n}\right]\cap\mathbb{Q}\Biggr)$ as a subset. Show that $S$ and $\mathbb{Q}^\mathbb{N}$ are equicardinal. – 2017-01-28
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0how intersection contains S , since S has uncountably many irrationals – 2017-01-28
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0I meant $\left[-\frac{1}{n},+\frac{1}{n}\right]\cap \mathbb{Q}$ (this should have been trivial to you). See the edit. – 2017-01-28
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0yes this works well – 2017-01-28
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1For every positive real number $r$, there is a square-summable rational sequence with square sum equal to $r$. – 2017-01-28
1 Answers
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Let $B$ be the set of binary sequences. Recall that $B$ and $\mathbb R$ have the same cardinality. The map $B \to \mathbb Q ^{\mathbb N} \cap l^2$ given by
$$(b_n) \to (b_1/1,b_2/2, \dots , b_n/n, \dots )$$
is injective, giving the result.