I've been struggling with this question I've been assigned for a couple weeks now.
The question is to show from ZFC that there is an uncountable sequence $〈f_α:α→ω:α<ω_1〉$ of functions such that
Each $f:α→ω$ is injective, and
for each $α<β<ω_1$, $\{ξ∈α:f_α(ξ)≠f_β(ξ)\}$ is finite.
I believe it will be a matter of constructing the sequence outright, but I can't seem to get the definition right so that it doesn't cause problems at limit ordinals, even if I avoid an infinite number of values in the range of $f_α$.
I may be thinking about this wrong, so any help would be appreciated.