A simple proof for the upper bound of a theorem of T. Łuczak

Let $b,c \gt 1$, and let $[a_1(x),a_2(x),a_3(x),\ldots ]$ be the continued fraction expansion of $x\in [0,1)$. The Hausdorff dimensions of the sets $$ \widetilde E(b,c)=\{x\in [0,1): a_n(x) \geq c^{b^n}\ \text{for all}\ n \in \mathbb N\} $$ and $$ E(b,c)=\{x\in [0,1): a_n(x) \geq c^{b^n}\ \text{for infinitely many}\ n \in \mathbb N\} $$ are equal to $1/(b+1)$, and play an important role in the dimension theory of continued fractions. A simple proof for the (optimal) lower bound of the Hausdorff dimension of $\widetilde E(b,c)$ was found by Feng et al. (1997). The proof for the upper bound given by Łuczak (1997) is based on an involved covering argument and a claim in combinatorics. In this note, we give a proof for the upper bound of the Hausdorff dimension of $E(b,c)$ without technical arguments.