(Global approximation by
smooth functions). Let
be a bounded set and
for
.
There exist a sequence
,
such
that

Proof

Let
Let
be a partition of unity subordinated to
.
Fix
.
Let
for
such
that

Let
Then

Proposition

(Approximation by smooth
functions). Let
be a bounded set,
admits a locally continuously differentiable parametrization and
for
.
There exist a sequence
,
such
that

Proof

1. For any point
there exists a radius
and a function
such
that

2.
Let
where the
is the
-th
coordinate vector. Let
.
Observe that

3. We introduce
and
claim

4. Let
are such
that
and
is such
that
Fix
.
According to the proposition
(
Local approximation by
smooth functions
) there exists a function
,
such
that
Let
be a smooth partition of unity subordinated to
.
We
set
We
claim
for some function
such
that