# Complex Metric Space

Consider the complex metric space $$\mathbb{C}$$ with the standard metric, let $$x\in \mathbb{C}$$. Show that if $$a\in B(x,r)$$ then there exists a non-zero h$$\in \mathbb{C}$$ such that $$a+h \in B(x,r)$$

May I have hints on how to prove this?

• Try using the triangle inequality. Oct 30, 2019 at 14:55
• @Gae.S. sorry, I forgot to say non-zero h.
– user643073
Oct 30, 2019 at 14:56

You can take $$h=\frac12\bigl(r-\lvert x-a\rvert\bigr)$$. That will work since, by the triangle inequality,$$B\bigl(a,r-\lvert x-a\rvert\bigr)\subset B(x,r).$$