Let
X
=
(
X
0
,
B
,
μ
,
T
)
X=(X^0,\mathcal {B},\mu ,T)
be an ergodic probability measure-preserving system. For a natural number
k
k
we consider the averages
(*)
1
N
∑
n
=
1
N
∏
j
=
1
k
f
j
(
T
a
j
n
x
)
\begin{equation*} \tag {*} \frac {1}{N}\sum _{n=1}^N \prod _{j=1}^k f_j(T^{a_jn}x) \end{equation*}
where
f
j
∈
L
∞
(
μ
)
f_j \in L^{\infty }(\mu )
, and
a
j
a_j
are integers. A factor of
X
X
is characteristic for averaging schemes of length
k
k
(or
k
k
-characteristic) if for any nonzero distinct integers
a
1
,
…
,
a
k
a_1,\ldots ,a_k
, the limiting
L
2
(
μ
)
L^2(\mu )
behavior of the averages in (*) is unaltered if we first project the functions
f
j
f_j
onto the factor. A factor of
X
X
is a
k
k
-universal characteristic factor (
k
k
-u.c.f.) if it is a
k
k
-characteristic factor, and a factor of any
k
k
-characteristic factor. We show that there exists a unique
k
k
-u.c.f., and it has the structure of a
(
k
−
1
)
(k-1)
-step nilsystem, more specifically an inverse limit of
(
k
−
1
)
(k-1)
-step nilflows. Using this we show that the averages in (*) converge in
L
2
(
μ
)
L^2(\mu )
. This provides an alternative proof to the one given by Host and Kra.