We show that every array
(
x
(
i
,
j
)
:
1
≤
i
>
j
>
∞
)
(x(i,j):1 \leq i > j > \infty )
of elements in a pointwise compact subset of the Baire-
1
1
functions on a Polish space, whose iterated pointwise limit
lim
i
lim
j
x
(
i
,
j
)
{\lim _i}{\lim _j}x(i,j)
exists, is converging Ramsey-uniformly. An array
(
x
(
i
,
j
)
i
>
j
)
(x{(i,j)_{i > j}})
in a Hausdorff space
T
T
is said to converge Ramsey-uniformly to some
x
x
in
T
T
, if every subsequence of the positive integers has a further subsequence
(
m
i
)
({m_i})
such that every open neighborhood
U
U
of
x
x
in
T
T
contains all elements
x
(
m
i
,
m
j
)
x({m_i},{m_j})
with
i
>
j
i > j
except for finitely many
i
i
.