0
$\begingroup$

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?

$\endgroup$
2
  • 1
    $\begingroup$ Try using the triangle inequality. $\endgroup$ Oct 30, 2019 at 14:55
  • $\begingroup$ @Gae.S. sorry, I forgot to say non-zero h. $\endgroup$
    – user643073
    Oct 30, 2019 at 14:56

1 Answer 1

2
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy