Q:

Norm inequality

While trying to compute a quotient space, the next problem has come to my attention:

Let $x=(x_j)_j$ and $y=(y_j)_j$ be two complex convergent sequences such that $x-y=(x_j-y_j)_j$, is a constant sequence.
  $$
\text{Note that $x-y$ is a constant sequence if and only if $x_j-x_{j+1}= y_j-y_{j+1}$ for each $j \in \mathbb{N}$.}
$$
  Consider the sequence $z=(y_j-y_{j+1})_j$. What is the order relation between $$
\| x \|_\infty = \sup_{j \in \mathbb{N}} | x_j | \ \ \text{ and } \ \ \| z \|_\infty= \sup_{j \in \mathbb{N}} | y_j - y_{j+1}| \ \ \text{ ?}
$$

My guess is that $\| x \|_\infty \geq \| z \|_\infty$. However, I can not prove it, so I am starting to think I may be wrong and no order relation can be obtained. My guess is not random, I have come up whit a few examples where it actually holds like the next ones:
$$
x=(1,1/2,1/3, \cdots ) \\
y=(1/2, 0 , -1/6 , \cdots)
$$
Clearly $x-y = (1/2, 1/2, 1/2, \cdots)$ and if $z=(y_j-y_{j+1})_j$ then 
$$
\| x \|_\infty=1 \geq 1/2=\| z \|_\infty
$$
Another example is obtained from the one above by changing $x$ and $y$ roles, that is since $y-x =(-1/2, - 1/2, - 1/2, \cdots)$, if now $z=(x_j-x_{j+1})_j$ then 
$$
\|y\|_\infty = 1/2 = \| z \|_\infty
$$
So, my question here is could someone help me on how to prove that my guess is correct or give a counterexample when $\|x\|_\infty < \| z \|_\infty$ ? I would appreciate it very much. Also if you think more details of the original problem can help let me know so I can make an edit. 

A:

Assuming I understand the question correctly, I believe it is false.
Consider $x=(50, -50, 50, -50, 0, 0, 0, 0, ...)$ and $y=(40, -60, 40, -60, -10, -10, -10,...)$.  Then $x-y=(10, 10, ...)$ and is constant.  $z=(-100, -100, -100, -70, -20, -20, -20, ...)$ so $\Vert z\Vert_{\infty} = 100$ and $\Vert x\Vert_{\infty} = 50$.

