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Does there exist a strictly increasing injective function $f \colon \omega _1 \mapsto \mathbb{R}$, where $\omega _1$ denotes the first uncountable ordinal?
No. If so, there is some rational between $f(\alpha)$ and $f(\alpha+1)$
for each $\alpha\in\omega_1$, and so uncountably many rationals, as these
rationals must be distinct.