Since Littlewood works in the 1960s, the boundedness of solutions of Duffing-type equations \(\ddot{x}+g(x)=p(t)\) has been extensively investigated. More recently, some researches have focused on the family of non-smooth forced oscillators \( \ddot{x}+\textrm{sign}(x)=p(t)\) , mainly because it represents a simple limit scenario of Duffing-type equations for when g is bounded. Here, we provide a simple proof for the boundedness of solutions of the non-smooth forced oscillator in the case that the forcing term p(t) is a T-periodic Lebesgue-integrable function with vanishing average. We reach this result by constructing a sequence of invariant tori whose union of their interiors covers all the \((t,x,\dot{x})\) -space, \((t,x,\dot{x})\in {\mathbb {S}}^1\times {\mathbb {R}}^2\) .
