= Solution
Choose $h_j\to0$. The assumed bound and the <weak subsequence of a bounded Hilbert-space sequence> result give a subsequence for which
$$
\Delta_i^{h_j}u\rightharpoonup g_i
\quad\hbox{weakly in }L^2(V),
\qquad
\lVert g_i\rVert_2\leq C.
$$
For every <test function> $\varphi\in C_c^\infty(V)$, a change of variables yields the difference-quotient integration-by-parts identity
$$
\int_V(\Delta_i^{h_j}u)\varphi
=-\int_Vu\,\Delta_i^{-h_j}\varphi.
$$
The right side converges to $-\int_VuD_i\varphi$, while the left side converges to $\int_Vg_i\varphi$. Thus $g_i$ is the $i$th <weak derivative> of $u$. This holds for every $i$, so
$$
\boxed{u\in H^1(V),\qquad
\lVert D_i u\rVert_{L^2(V)}\leq C.}
$$
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