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?
2026-05-05 05:47:03.1777960023
Complex Metric Space
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You can take $h=\frac12\bigl(r-\lvert x-a\rvert\bigr)$. That will work since, by the triangle inequality,$$B\bigl(a,r-\lvert x-a\rvert\bigr)\subset B(x,r).$$