= Solution
Provided the limit belongs to the original admissible class, the definition of the <infimum> and the assumed inequality give
$$
\boxed{l\leq I(u)\leq\liminf_k I(u_{j_k})=l,\qquad I(u)=l.}
$$
The 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>.
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