For $ x\in[0,1), $ let $ [d_{1}(x),d_{2}(x),\ldots] $ be its Lüroth expansion and $ \big\{\frac{p_{n}(x)}{q_{n}(x)}, n\geq 1\big\} $ be the sequence of convergents of $ x. $ In this paper, we study the Jarník-like set of real numbers which can be well approximated by infinitely many of their convergents in the Lüroth expansion$ \colon $
$ W(\psi) = \{x\in[0,1)\colon |xq_{n}(x)-p_{n}(x)|<\psi(n) \text{ for infinitely many } n\in \mathbb{N}\}, $
where $ \psi\colon \mathbb{R}\to (0,\frac{1}{2}] $ is a positive function. We completely determine the Hausdorff dimension of $ W(\psi). $
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