Given that $a, b$ and $c$ are three real numbers, so we can take $a = 1,b = 2$ and $c = 3$
Now verify each option one by one.
A.$\mid a+b \mid\leq \mid a \mid + \mid b \mid$
$\implies \mid 1 + 2 \mid\leq \mid 1 \mid + \mid 2 \mid$
$\implies 3 \leq 3\:\text{(True)}$
B.$\mid a – b \mid \leq \mid a \mid + \mid b\mid$
$\implies\mid 1 – 2 \mid \leq \mid 1 \mid + \mid 2 \mid$
$\implies 1 \leq 3\:\text{(True)}$
C.$\mid a – b \mid \leq \mid a \mid – \mid b \mid$
$\implies \mid 1 – 2 \mid \leq \mid 1 \mid – \mid 2 \mid$
$\implies 1 \leq -1\:\textbf{(False)}$
D.$\mid a-c \mid \leq \mid a-b \mid+\mid b-c \mid$
$\implies \mid 1-3 \mid \leq \mid 1-2 \mid+\mid 2-3 \mid$
$\implies \mid -2 \mid \leq \mid -1 \mid+\mid -1 \mid$
$\implies 2 \leq 2\:\text{(True)}$
So, the correct answer is $(C).$