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We prove that in any sequence of distinct modular curves over k, the k-gonality tends to infinity. This extends earlier work, in which the result was proved for particular sequences of modular curves, such as X_0(N) for p not dividing N. 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Let X(p^e;N) be the curve parameterizing elliptic curves with full level N structure (where p does not divide N) and full level p^e Igusa structure. By modular curve, we mean a quotient of any X(p^e;N) by any subgroup of ((Z/p^e Z)^* x \\SL_2(Z/NZ))/{+-1}. We prove that in any sequence of distinct modular curves over k, the k-gonality tends to infinity. This extends earlier work, in which the result was proved for particular sequences of modular curves, such as X_0(N) for p not dividing N. 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