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Provided the limit belongs to the original admissible class, the definition of the infimum and the assumed inequality giveThe needed fundamental property is sequential lower semicontinuity for the convergence in the compactness argument: uniform, or strong $L^1$, convergence of functions together with weak $L^1$ convergence of derivatives. The extra integrand hypothesis in the classical Tonelli existence theorem is convexity in the derivative variable $v$ for every fixed $(x,u)$, together with the stated coercivity and continuity. In one dimension the resulting integral is lower semicontinuous under the uniform/weak-derivative convergence above.

This lower-semicontinuity assertion does not itself supply the missing Lipschitz membership in (iii). With the natural fixed-trace $W^{1,1}$ admissible class and its own infimum, it completes Tonelli's direct existence argument. Retaining the printed Lipschitz class requires the additional membership estimate; changing admissible classes can otherwise encounter the Lavrentiev phenomenon.