Our method from converting information about the tubes into information about M will involve pulling back a test function on M to each of the tubes. The functions on the tubes that we will obtain in this way will not necessarily be smooth, because we are allowing the tubes to zig-zag. It would be possible to work only with cuttings into smooth tubes, but we prefer to allow ourselves the extra latitude in cutting, and pay the price by smoothing the pulled-back test functions, rather than the tubes themselves.
Lemma.
If 
is piecewise 
,
has compact support, and doesn't vanish identically,
then
Proof.
Choose 
to be smooth,
with support in the interval [-1,1] and
,
and convolve 
with 
.
The result is a member of 
whose Rayleigh quotient approaches that of as 
.
qed