I have been looking a little closer at the following:
There exists a linear homeomorphism $T:(c, \|$ $\left.\|_{\infty}\right)
\rightarrow\left(c_o,\|\cdot\|_{\infty}\right)$, where $c$ and
$c_{0}$ refer to the sequence spaces of convergent sequences/sequences that
converge to zero, respectively.
I am aware that there has also been a discussion regarding whether or not these spaces not being isometric to one another, such as here https://mathoverflow.net/questions/300108/c-0-is-not-isometrically-isomorphic-to-c
Also, a discussion regarding them being isomorphic can be found here Is $c$ isomorphic to $c_0$?
-> However, I found a (very) recent paper addressing both problems in a fairly elementary manner, which can be found here:
https://link.springer.com/article/10.1007/s12045-023-1588-2
and I am studying it.
The one and only inequality I could not grasp shows up in Theorem 1, page 3 of the paper, where it is stated that:
$$
\text { for } n \in \mathbb{N}, \left|x_n\right|=\left|x_n-x_{\infty}+x_{\infty}\right| \leq\left|x_n-x_{\infty}\right|+\left|x_{\infty}\right| \leq 2\|T x\|_{\infty}
$$
Why does the last inequality holds?
It really is a silly doubt, so I am probably missing something simple, but I would really like to know why this is true, as I am checking each detail carefully.
Observation: This paper is very interesting, at least for me, and probably for more people that have been looking at this problem, as it summarizes some results and proves them in an elementary functional analysis setting.
Observation 2:
In the proof, the map $T: c \rightarrow c_o$ is defined as:
$$
T(x)=T\left(\left(x_n\right)\right)=T\left(x_1, x_2, \ldots, x_n, \ldots\right)=\left(x_{\infty}, x_1-x_{\infty}, x_2-x_{\infty}, \ldots\right)
$$
where $x_{\infty}=\lim _{n \rightarrow \infty} x_n$
Best Answer
$\|Tx\|=\sup \{|x_{\infty}, |x_1-x_{\infty}|,|x_2-x_{\infty}|,\cdots\}$. Supremum of a set of real numbers is greater than or equal to each member of the set. So $\|Tx\| \geq |x_{\infty}|$ and $\|Tx\| \geq |x_n-x_{\infty}|$ for any $n$. Add these two inequalities.