I am trying to find the Hausdorff dimension of $F=\{1,\frac{1}{4},\frac{1}{9},\ldots\}\cup\{0\}$. This question: Hausdorff Dimension Calculation gives some help but I don't think it finishes the answer.
It gives that: there are $k+1$ points, each of which can be covered by an interval of size $\delta=k^{-2}$. So $$\mathcal{H}^{s}(F)\leq\lim_{k\to\infty}\frac{k+1}{k^{2s}}.$$
I think that this gives $\dim_{\text{H}}(F)=\frac{1}{2}$ since $\frac{k+1}{k^{2s}}=\frac{k}{k^{2s}}+\frac{1}{k^{2s}}$ which is bounded (goes to $1$) for all $s\geq\frac{1}{2}$, but tends to $\infty$ for all $s<\frac{1}{2}$, and tends to $0$ for all $s>\frac{1}{2}$.
However, I'm not sure how to write this out formally. Please help.