= Solution
Write $l=\inf_{\mathcal A}I$. Since the limit is admissible, the assumed lower-semicontinuity inequality gives $l\leq I(u)\leq\liminf I(u_{j_k})=l$. Thus \b[$I(u)=l$ and $u$ is a <global minimizer>].
The extra integrand hypothesis in the <Tonelli existence theorem> for one dimension is \b[<convexity> in the derivative variable for each fixed $(x,u)$]. Together with the stated <continuity> and coercive lower bound it gives the lower-semicontinuity property along the uniform-<function>/weak-derivative convergence just proved.
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