= Uncountability by interleaving prescribed binary coordinates
Partition the positive <natural numbers> into three infinite coordinate <sets>. Fix the coordinates on the first two <sets> to any prescribed binary values, and leave those on the third <set> free. Reading the third coordinate <set> embeds all infinite <binary sequences> in the resulting family, so the family is an <uncountable set> by <Cantor's diagonal argument>. For example, fixing $x_{3k-2}=0$ and $x_{3k-1}=a_{3k-1}$ while leaving $x_{3k}$ free guarantees infinitely many zeros and infinitely many agreements with any prescribed <binary sequence> $(a_n)$. Infinite free coordinates remain even after both requirements are enforced.
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