Let 
and 
be elements of the image
we wish to connect by a path.
Find 
, 
such that
have the same first coordinate,
and 
, 
have the same first coordinate and
.
(If 
have the same first coordinate, take 
;
otherwise apply the hypothesis 
with v as the empty vector.)
Let 
be this common value.
Next find 
, 
,
using the same argument, such that 
agree in the first two
coordinates,
, 
agree in the first two coordinates, and
.
Let 
be this common value.
Do the analogous thing at 3/4.
By induction, we may continue to define 
, 
,
at all dyadic rationals
in 
, such that
and 
agree in the first n coordinates
and
By induction, we see that all the 
with
agree in the first n coordinates.
Hence for
not a dyadic rational, we may define
and 
.
Then w maps
into 
where
is the first n coordinates of 
,
, and of
.
We now show that w is continuous at 
.
Let U be an open set of Y containing 
.
Then 
contains 
and 
for some finite
substrings 
of 
respectively, by continuity of f.
By the last sentence of the previous paragraph it follows that
will contain a neighborhood of r.
Thus 
is a continuous
path, and image 
is path connected. 
