I want to construct a sequence of functions $$f\_\alpha: \alpha \rightarrow \omega,\ \alpha < \omega\_1$$
such that for all $\alpha < \omega\_1$ the following holds:


* $f\_\alpha$ is injective.
* $(\forall \beta < \alpha)\ f\_\beta =^\* f\_\alpha \upharpoonright \beta$,


i.e. the restriction $f\_\alpha \upharpoonright \beta$ agrees with $f\_\beta$ in all but finitely many places.


I feel this should be possible for similar reasons we can construct a family of continuum many almost disjoint subsets of $\omega$.



Constructing an $\omega_1$-sequence of functions that almost extend all previous functions
Nice question! I like it very much. 


Sure, we can do this. We'll also arrange that
$\text{ran}(f\_\alpha)$ is coinfinite. That will make the successor
steps easy, since there is always another point available. The
difficulty is what to do at limits. Suppose we have $f\_\alpha$ for
$\alpha<\lambda$, where $\lambda$ is a countable limit ordinal. We want to define the function $f\_\lambda$. In order to do this, pick an increasing cofinal sequence
$\alpha\_0<\alpha\_1<\alpha\_2<\cdots<\lambda$ with
$\sup\_n\alpha\_n=\lambda$. We shall define $f\_\lambda$ in blocks. First, let $f\_\lambda$ agree with
$f\_{\alpha\_0}$ up to $\alpha\_0$. Next, on the interval
$[\alpha\_0,\alpha\_1)$, we make $f\_\lambda$ agree with
$f\_{\alpha\_1}$, except modified so that it is still injective in combination with what we already did below $\alpha\_0$.
This modification will require only at most finitely many changes to $f\_{\alpha\_1}\upharpoonright[\alpha\_0,\alpha\_1)$, since
$f\_{\alpha\_0}$ and $f\_{\alpha\_1}$ agree except on a finite set.
Continuing, we define $f\_\lambda$ on the interval
$[\alpha\_n,\alpha\_{n+1})$ to agree with $f\_{\alpha\_{n+1}}$, except
fixed up again to be injective, which will require only finitely many
changes. Ultimately, in this way we'll get an injective function
$f\_\lambda:\lambda\to\omega$. For any $\alpha<\lambda$, we have
$\alpha<\alpha\_n$ for some $n$, and consequently
$f\_\lambda\upharpoonright\alpha$ was obtained from finite
modifications of the $f\_{\alpha\_k}$'s for $k<n$, each of which had
only finite difference from $f\_\alpha$ up to $\alpha$. And so our
function $f\_\lambda$ differs from $f\_\alpha$ on $\alpha$ only at
most finitely. Lastly, to ensure that $f\_\lambda$ has coinfinite
range, we may if necessary simply change the values on the
ordinals $\alpha\_n$, since this will involve only finitely many
changes below any given $\alpha<\lambda$. Thus, we have constructed
the function $f\_\lambda$ with the desired properties, the
construction may proceed up to $\omega\_1$. 


