- #1

- 317

- 25

## Homework Statement

Let ##S\subseteq \Bbb{R}## and ##T = \{ t\in \Bbb{R} : \exists s\in S, \vert t-s\vert \lt \epsilon\}## where ##\epsilon## is fixed. I need to show T is an open set.

## Homework Equations

n/a

## The Attempt at a Solution

Let ##x \in T##, then ##\exists \sigma \in S## such that ##x \in (\sigma -\epsilon, \sigma +\epsilon)##. Let ##\delta = \min(\frac{\vert \sigma -\epsilon -x\vert}{2}),\frac{\vert \sigma +\epsilon -x\vert}{2}## and consider the interval ##(x-\delta, x+\delta)##. If ##y \in (x-\delta, x+\delta)##, then

\begin{align}

\vert \sigma - y \vert \leq \vert \sigma -x\vert + \vert x- y\vert \lt \epsilon + \delta \cdots

\end{align}

I want to show ##\vert \sigma -y \vert \lt \epsilon##, but I am having trouble with finding the right expression to simplify this