(Solution) - Let R a b c d be a -(2025 Original AI-Free Solution)

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Let R = [a, b] × [c, d] be a rectangle in R2. A function f is said to have separated variables if

for some scalars ck and functions fk ? C[a, b], gk ? C[c, d]. Prove that given f ? C(R) there is a sequence of functions with separated variables, Pn, such that Pn ? f uniformly on R as n ? ?.