  Let $\{s_n\}$ and $\{t_n\}$ be two sequences of positive real numbers. Under
some mild conditions on $\{s_n\}$ and $\{t_n\}$, we give the precise formula of
the Hausdorff dimension of the set \[
\mathbb{E}(\{s_n\},\{t_n\}):=\Big\{x\in(0,1): s_{n}<d_{n}(x)\leq s_n+t_n,
\forall n\geq1\Big\}, \] where $d_n(x)$ denotes the digit of the Engel
expansion of $x$. This result improves the Lemma 2.6 of Shang and Wu (2021JNT),
and is very useful for calculating the Hausdorff dimension of certain sets in
Engel expansions.
