Let
X
X
be a Stein manifold of dimension
n
≥
1
n\ge 1
. Given a continuous positive increasing function
h
h
on
R
+
=
[
0
,
∞
)
\mathbb {R}_+=[0,\infty )
with
lim
t
→
∞
h
(
t
)
=
∞
\lim _{t\to \infty } h(t)=\infty
, we construct a proper holomorphic embedding
f
=
(
z
,
w
)
:
X
↪
C
n
+
1
×
C
n
f=(z,w):X\hookrightarrow \mathbb {C}^{n+1}\times \mathbb {C}^n
satisfying
|
w
(
x
)
|
>
h
(
|
z
(
x
)
|
)
|w(x)|>h(|z(x)|)
for all
x
∈
X
x\in X
. In particular,
f
f
may be chosen such that its limit set at infinity is a linearly embedded copy of
C
P
n
\mathbb {CP}^n
in
C
P
2
n
\mathbb {CP}^{2n}
.