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We present a sequent calculus for the Grzegorczyk modal logic 
$\mathsf {Grz}$
allowing cyclic and other non-well-founded proofs and obtain the cut-elimination theorem for it by constructing a continuous cut-elimination mapping acting on these proofs. As an application, we establish the Lyndon interpolation property for the logic 
$\mathsf {Grz}$
proof-theoretically.
