I've been working on determining the VC-dimension of a specific family of classifiers, and I would like to get some feedback on the proof I've come up with. The family of classifiers is defined as follows: $$ \mathcal{H} = \{h_{a,b} : a, b \in \mathbb{R} \,|\, h_{a,b}(x, y) = \begin{cases} 1, & \text{if } x \geq a \text{ and } y \geq b, \\ 0, & \text{else}. \end{cases} $$
I've provided a proof for the VC-dimension of this class, and I would appreciate it if someone could check its correctness or suggest improvements. Here's the proof:
Find the VC-dimension of this class with a full proof.
\subsection*{Definition of Shattering:} In order to find the VC-dimension, we are looking for the largest set of points that can be shattered by the classifiers in $\mathcal{H}$. A set of points is shattered if, for every possible labeling of the points (assigning 0 or 1 to each point), there exists a classifier $h_{a,b}$ in $\mathcal{H}$ that produces that particular labeling.
Finding VC-dimension of $\mathcal{H}$
To find the VC-dimension of the given class of classifiers $\mathcal{H}$, we need to determine the maximum size of a shattered set. Let's analyze the class of classifiers given by equation~\ref{eq:classifier}. Let's consider a set of points $\{(x_1, y_1), (x_2, y_2), \ldots, (x_m, y_m)\}$ and check the number of distinct labelings we can achieve with the classifiers in $\mathcal{H}$.
For each point $(x_i, y_i)$, the classifier $h_{a,b}$ assigns a label of 1 if $x_i \geq a$ and $y_i \geq b$, and a label of 0 otherwise.
Now, consider a set of $m$ points $\{(x_1, y_1), (x_2, y_2), \ldots, (x_m, y_m)\}$. For each point, there are two possible labelings (0 or 1), so there are $2^m$ possible labelings for the entire set.
To find the VC-dimension, we need to find the largest $m$ such that $\Pi_{\mathcal{H}}(m) = 2^m$. This means we want to find the largest set of $m$ points that can be shattered by the classifiers in $\mathcal{H}$.
For the given class $\mathcal{H}$, it can shatter any set of $m$ points because we can always find $a$ and $b$ such that the labels assigned by $h_{a,b}$ match any labeling of the points. Therefore, $\Pi_{\mathcal{H}}(m) = 2^m$ for any $m$.
So, the VC-dimension of $\mathcal{H}$ is infinite: $VC_{\text{dim}}(\mathcal{H}) = \infty$.
Given any set of $m$ points, we can find $a$ and $b$ values such that the classifiers in $\mathcal{H}$ produce any possible labeling of the points. This is because we have the flexibility to adjust $a$ and $b$ to meet the conditions $x \geq a$ and $y \geq b$ for any combination of points. Therefore, $\Pi_{\mathcal{H}}(m) = 2^m$, meaning that the class $\mathcal{H}$ can shatter any set of $m$ points.
The VC-dimension is the size of the largest shattered set. Since $\Pi_{\mathcal{H}}(m) = 2^m$ for any $m$, there is no largest $m$. In this case, $VC_{\text{dim}}(\mathcal{H}) = \infty$ because $\mathcal{H}$ can shatter sets of any size.$$
Here is also the image form if you want to read more clearly: 
Any feedback, corrections, or suggestions on how to make the proof more robust would be highly valuable. Thank you!
