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Since Littlewood works in the 1960's, the boundedness of solutions ofDuffing-type equations $\ddot{x}+g(x)=p(t)$ has been extensively investigated.More recently, some researches have focused on the family of non-smooth forcedoscillators $ \ddot{x}+\text{sgn}(x)=p(t)$, mainly because it represents asimple 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-smoothforced oscillator in the case that the forcing term $p(t)$ is a $T$-periodicLebesgue integrable function with vanishing average. We reach this result byconstructing a sequence of invariant tori whose union of their interiors coversall the $(t,x,\dot x)$-space, $(t,x,\dot{x})\in\mathbb{S}^1\times\mathbb{R}^2$.

Invariant tori and boundedness of solutions of non-smooth oscillators with Lebesgue integrable forcing term