I am having trouble completely understanding the mathematical definition of non-satiation. I have stated the definition from Wikipedia below. It would be great if someone can graphically explain.
Formally, if $X$ is the consumption set, then for any $x \in X$ and every $\varepsilon > 0$, there exists $y \in X$ such that $||y-x|| \leq \varepsilon$ and $y$ is preferred to $x$.
The Wikipedia article shows it graphically. But here, you're leaving out the important bit: "The property of local nonsatiation of consumer preferences states that for any bundle of goods there is always another bundle of goods arbitrarily close that is preferred to it." This is the statement that you want to connect with the math.
So how does this relate? Let's go bit-by-bit. Refer to the graph on the Wikipedia page as needed.
-"for any bundle of goods" means "choose any bundle $x \in X$, where X is the box in the Wikipedia graph.
-"arbitrarily close" means "choose any $\epsilon > 0, \epsilon \in \mathbb{R}$". So, pick any non-zero real number as small or as large as you like.
-"there is always another bundle of goods arbitrarily close" means "given $x$ and $\epsilon$, you can always find a bundle $y \in X$ that is within a distance $\epsilon$ from $x%$." This is expressed as $||y-x|| \le \epsilon$.
-"is preferred to it" means that for any utility function $U(x)$, we have $U(y) > U(x)$.
Putting this all together, "non-satiation" means that (without a budget constraint) an agent can "never be satisfied" because no matter what bundle of goods they have ($x$) there is always a $y$ that yields greater utility. "Local" means that $y$ and $x$ can be very similar (ie arbitrarily close to each other in $X$).
