Using the Schröder–Bernstein theorem show that (0,1) × (0,1) and (0,1) have the same cardinality.
I've just recently been introduced to cardinality and have been struggling with this question.
2026-04-07 07:53:16.1775548396
Schröder–Bernstein Theorem
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The theorem requires you to inject each way. Injecting from one dimension to two dimensions is trivial (just add a coordinate equal to $\frac{1}{2}$). But how do we do the reverse? Here's a hint: each number in $\left( 0,\,1\right)$ has a unique decimal expansion, provided you ban infinite trailing $9$s. How can I inject from pairs of these expansions to such expansions singly?