Comparing the two cardinals $\aleph_0^{\aleph_0}$ and $2^{\aleph_0}$ [duplicate]

set-theory

$\aleph_0^{\aleph_0}\le\left(2^{\aleph_0}\right)^{\aleph_0}=2^{\aleph_0\cdot\aleph_0}=2^{\aleph_0}\le\aleph_0^{\aleph_0}$.

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