Let \(f\colon\mathbb{N}\to\{0,1\}^{\mathbb{N}}\) be any function. Which of these sequences is guaranteed not to lie in the image of \(f\), for every choice of \(f\)?
\(u(n)=1-f(n)(n)\) differs from \(f(n)\) at position \(n\) for every \(n\), so it never equals any \(f(n)\). The other options may coincide with some \(f(k)\) depending on the chosen function.