Proving a rectangle is connected.

[gottfried]

Homework Statement

Let K ={(x,y)[itex]\in[/itex]ℝ²:|x|≤1,|y|≤1}
Prove that K is a connected subset of ℝ²

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[itex]\in[/itex]f(x),b[itex]\in[/itex]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?

 

[mfb]

f(x)={(x,y):|y|≤1}

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.