# Proving a rectangle is connected.

1. May 5, 2013

### gottfried

1. The problem statement, all variables and given/known data
Let K ={(x,y)$\in$ℝ2:|x|≤1,|y|≤1}
Prove that K is a connected subset of ℝ2

3. The attempt at a solution

Suppose f:[-2,2]→K and define f(x)={(x,y):|y|≤1}

Dist(f(x),f(y))=sup(d(a,b):a$\in$f(x),b$\in$f(y))=d(x,y)=|x-y|. Using this equality it is easily shown that f(x) is continuous.

So f is continuous and [-2,2] is connected therefore f([-2,2])=k is connected.

Proving things are connected is very difficult and was just wondering if my proof was vaguely correct?

2. May 5, 2013

### Staff: Mentor

What does that mean? What is f(0), for example?

Here is a possible approach: for every two points (x1,y1), (x2,y2) in K, the straight line between them is part of K.