I'm currently self-studying Hubbard and Hubbard's "Vector Calculus" and trying to do all the exercises. Currently I'm working on 0.4.4. I think I feel intuitively what is trying to be achieved, but I can't understand the notation. The function in question is:
$$\tilde{f}(\mathbf{x}) = \sup_{k} \inf_{l \geq k} f([x_{1}]_{l}, \ldots, [x_{n}]_{l})$$
Where $[x]_n$ denotes the portion of some real number $x$ that can be expressed with all the digits before the decimal and $n$ digits after the decimal, and $D$ is the set of finite-decimal numbers. I'm trying to show that this function has certain properties, but I can't parse the notation describing the function. Specifically, I'm confused about how the "output" of the inf operator can be a set, since k is fixed arbitrarily.
Any clarification would be greatly appreciated.
Best Answer
I interpret the notation $$\sup_{k}$$ as meaning $$\sup_{k \in \mathbb{N}}.$$ In other words, we are considering the supremum over all natural numbers $k$.
So, $$\tilde{f}(\mathbf{x}) = \sup_{k} \inf_{l \geq k} f([x_{1}]_{l}, \ldots, [x_{n}]_{l})$$ would mean:
First, fix some arbitrary $k_{1} \in \mathbb{N}$. Then, for each $l \geq k_{1},$ calculate $$f([x_{1}]_{l}, \ldots, [x_{n}]_{l}).$$ This gives you a set of values $$S_{k_{1}} = \{f([x_{1}]_{l}, \ldots, [x_{n}]_{l}) \mid l \geq k_{1}\}.$$ Then, take the infimum of this set, and call it $I_{k_{1}}$; in other words, $$I_{k_{1}} = \inf S_{k_{1}}.$$
Finally, since $k_{1}$ was arbitrary, we get a $S_{k}$ and corresponding infimum $I_{k}$ for every $k \in \mathbb{N}$. Our desired value of the function is then $$\tilde{f}(\mathbf{x}) = \sup \{I_{k} \mid k \in \mathbb{N}\}.$$