I'm trying to, for the purpose of understanding Rudin's definition of boundedness in a metric space, reconcile this with the definition of boundedness in $\mathbb{R}^n$ I'm accustomed to. Here are the two definitions. Rudin's definition is for a general metric space, but I'm going to limit this to just $\mathbb{R}^n$ for the moment.
(1) My "Intuitive" Definition
In $\mathbb{R}^n$ with the Euclidean metric, we say $E \subset \mathbb{R}^n$ is bounded if there exists $M > 0$ such that for every $x \in E$, we have $|x| < M$.
(2) Rudin's definition (Definition 2.18(i))
$E$ is bounded if there is a real number $M$ and a point $p \in X$ such that $d(p,q) < M$ for all $p \in E$.
These definitions appear very different to me, and the second is a bit unintuitive. Here is my attempt to reconcile them. If $E \subset \mathbb{R}^n$ is bounded by the first definition, then it is also bounded by the second definition by taking $p$ to be the zero vector. The other direction I'm less sure on. Let's saying that $E$ is bounded by definition (2). Then there exists $M$ (I might as well assume $M > 0$, since $d$ is non-negative anyway) and $p \in \mathbb{R}^n$ such that for every $x \in \mathbb{R}^n$, we have $d(p,x) = |p-x| < M$. Given $x \in \mathbb{R}^n$, I can attempt to bound $|x|$ by the triangle inequality: $$ |x| = |(x-p) + p| \leq |x-p| + |p| < M + |p|. $$ I think, since this $p$ is fixed at the moment, I can choose a bound that depends on $p$. So I can set $N = M + |p|$. Then, for every $x \in \mathbb{R}^n$, we have $|x| < N$.
How does this look? If there is a tighter bound that I'm missing, I'd appreciate some help with finding it. Also, any intuition on why this second definition makes sense would be very helpful.