proof $\lim x_n \le \lim y_n$ if $x_n \le y_n$ (clarification)

Question

In this proof, I don't understand the statement that 'for some $n \ge \max\{M_1,M_2\}$, we have $x-x_n<\varepsilon/2$.' How can $|x_n-x|$ be changed into $x-x_n$??

Thanks you in advance.

Answer 1

Recall that $|a| \leq b$ is equivalent to $-b \leq a \leq b$. Therefore $|x_n - x| < \epsilon/2$ gives you $-\epsilon/2 < x_n - x < \epsilon/2$.

Answer 2

Notice that $$x-x_n \le |x-x_n| <\varepsilon/2$$

Similarly $$y-y_n \le |y-y_n| <\varepsilon/2$$