Does $| a - b | \geq a - b $ always hold for any $a, b \in \mathbb{R} $?

Question

I saw this relation in a proof involving convergence of real sequences (if you're interested, I can give you the details, but I reckon they're not essential for answering my question):

$| x_n - x^* | \geq x_n - x^* $, for all $n > N$ and $N \in \mathbb{N} $, where $x_n$ is a real sequence and $x^*$ its limit.

I guess my question boils down to whether $| a - b | \geq a - b $ always holds for any $a, b \in \mathbb{R} $.

Intuitively and plugging in different combinations of positive and negative numbers, this seems to hold. However, I haven't seen this relation listed as a property of the absolute value anywhere.

Can you confirm this relation always holds and am I safe to use it?

Answer 1

For any real number $r$, $|r|\geqslant r$. So, if $a,b\in\Bbb R$, $|a-b|\geqslant a-b$.