This may seem like a tricky question at first because there may seem to be not enough "space" to fit
into
, but actually there is more "space" than you think. Let
be a sequence of distinct elements contained in the interval
where
and let
. Define the function
,
\begin{align*}
f(x) = \begin{cases} x_{n+1} & \text{if \ } x \in M \\ x & \text{if \ } x \in [0,1] \setminus M.\end{cases}
\end{align*}
To show that
is injective, we consider the following three cases:
1) Let
. Note that
and
are both elements of
. Since
only consists of distinct elements, then
implies
.
2) Let
,
. Note that
and
. Thus, if
then
.
3) Let
. Note that
and
, so if
then
.
To show that
is surjective, we need to show that the range of
is equal to
. The range of the function
,
, is
and the range of the function
,
, is
. Since
is a piecewise function of
and
, its range must be precisely
and we are done.
EDIT: Don't know what's up with AN's latex typeset, it's way different and very non-standard from back in the days I used it lol.