Proving a metric is continuous

[radou]

Homework Statement

So, given a metric d : X x X --> R, prove that d is continuous.

The Attempt at a Solution

Let (x, y) be a point in X x X, V = <a, b> a neighborhood of d(x, y). One needs to find a neighborhood of U of (x, y) such that d(U) is contained in V. U is of the form U1 x U2, where U1 is a neighborhood of x, and U2 a neighborhood of y. I claim that every union U of two open balls B1(x, r1) and B2(y, r2), where 2(r1 + r2) = b - a, must satisfy d(U) $$\subseteq$$ <a, b>.

The diameter of B1 is 2r1, and the diameter of B2 is 2r2. The diameter of their union is b = a + 2r1 + 2r2, where a is the distance between B1 and B2.

Does this work?

 

[mighty2000]

Yes, your proof is correct. You have correctly identified the neighborhoods U1 and U2 that contain x and y respectively, and have shown that their union U also satisfies the condition of being a neighborhood of (x, y). Additionally, you have shown that the diameter of U is equal to the distance between the neighborhoods V and <a, b>, which proves that d(U) is contained in V. Therefore, d is continuous. Great job!