this paper theorem $7.$
Theorem $7$. Let $(X,U)$ be a boundedly compact uniform space. Then the mapping $θ_I :(bs(X),U˜ ) → (K(X),U_H )$ is uniformly continuous.
Proof: Let $U ∈ U$. Choose a $V ∈ U$ such that $V^3 ⊂ U.$ Let $((x_n), (y_n)) ∈ V˜$ and $p ∈ I(C_x)$ (where $x = (x_n)_{n∈N})$. Then $B = \{n ∈ N: x_n ∈ V [p]\} \not∈ I$. Now for each $n ∈ B, (x_n, y_n) ∈ V$ and $(x_n, p) ∈ V$ so that $(y_n, p) ∈ V^ 2$, i.e. $y_n ∈ V^ 2[p]$. Hence
$\{n ∈ N: y_n ∈ V^2[p]\}⊃\{n ∈ N: x_n ∈ V [p]\}$
which implies that the set on the left hand side also cannot belong to $I$. Again as $V^2[p]$ is compact, by Theorem $6$, $V^ 2[p] ∩ I(C_y ) = ∅ (y = (y_n)_{n∈N})$. This implies that $p ∈ V^3[I(C_y)]$. Therefore $$I(C_x) ⊂ V^3[I(C_y)]⊂U[I(C_y)].$$
Similarly it can be shown that $I(C_y ) ⊂ U[I(C_x)]$. Hence $(I(C_x),I(C_y)) ∈ U ∈ U_H$.
The point I have a problem is that I don't know what is meant by $V^3[I(C_y)]$ or $U[I(C_x)].$ If anybody explains that to me, may bay be I can do the rest. thank you.