The authors prove that a singular cardinal κ can be extended to include an unbounded sequence without adding bounded subsets or altering κ^+. This solves Solovay’s problem (SP) for any cofinality of κ, provided κ is a strong limit when the cofinality is uncountable. They define a forcing notion P̄κ[¯F] built from trees of height ω (or θ for uncountable cofinality) whose branching is governed by a sequence of filters. The construction guarantees that P̄κ[¯F] adds a θ‑sequence to κ, preserves κ^+, and does not introduce any bounded sequences. The method relies on κ being a strong limit when cf(κ) > ℵ₀; without this assumption the argument fails for countable cofinality. The result extends earlier partial solutions and provides a complete resolution of SP.

Fact: P̄κ[¯F] is defined using trees of height ω whose splitting nodes are controlled by a sequence ¯F of filters. The forcing adds an ω‑sequence to κ and preserves κ^+ while adding no bounded subsets.

Fact: For uncountable cofinality, the same construction applied to a tree of height θ (where θ = cf(κ)) yields analogous results.

Fact: The methods require κ to be a strong limit cardinal when cf(κ)>ℵ₀; without this assumption the construction does not produce (SP) in the countable cofinality case.

Fact: The result settles Solovay’s problem, allowing the addition of an unbounded sequence to a singular cardinal without affecting its successor or creating bounded subsets.