Math CentralQuandaries & Queries


Question from Aqeel, a teacher:

Theorem: The open ball is an open set?


Draw a diagram!

open ball

My diagram is the open ball of radius $r$ about the point $p.$ Let $x$ be some point in this ball so the distance from $p$ to $x$ is less than $r.$ Construct an open ball about $x$ of radius $\epsilon$ so that the ball remains inside the ball about $p$ of radius $r.$ What can $\epsilon$ be?

Now write a formal proof.


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