Self-Study: Triangle Inequality Correct?

Question

I am presented the following inequality for $f:D\rightarrow \Bbb{C}$: $$|f(x)| \geq |f(\hat{x})| - |f(x) - f(\hat{x})|$$

My explanation is:

$$|f(\hat{x})| = |f(\hat{x}) - f(x) + f(x)| = |(f(\hat{x}) - f(x)) + f(x)| \leq |(f(\hat{x}) - f(x))| + |f(x)|$$

Thus: $$|f(\hat{x})| - |(f(\hat{x}) - f(x)| \leq |f(x)|$$

Since $|x|= |-x|$ it follows: $$|f(\hat{x})| - |(f(\hat{x}) - f(x)| = |f(\hat{x})| - |(f(x) - f(\hat{x})| \leq |f(x)|$$

is my reasoning correct?

Answer 1

Your reasoning is correct. ${}{}{}{}{}{}{}{}$