Metric Spaces, Triangle Inequality

[cwatki14]

I have the following question on metric spaces

Let (X,d) be a metric space and x1,x2,...,xn ∈ X. Show that
d(x1, xn) ≤ d(x1, x2) + · · · + d(xn−1, xn2 ),
and
d(x1, x3) ≥ |d(x1, x2) − d(x2, x3)|.

So the first part is simply a statement of the triangle inequality. However, the metric isn't given. How do I prove the triangle equality for a metric space when the metric isn't given?

For the second part, I attempted a few solns using a typical approach with the triangle inequality, however none of them seemed to be working... Any ideas?

 

[vela]

I have the following question on metric spaces

Let (X,d) be a metric space and x1,x2,...,xn ∈ X. Show that
d(x1, xn) ≤ d(x1, x2) + · · · + d(xn−1, xn2 ),
and
d(x1, x3) ≥ |d(x1, x2) − d(x2, x3)|.

So the first part is simply a statement of the triangle inequality. However, the metric isn't given. How do I prove the triangle equality for a metric space when the metric isn't given?

It's not exactly a statement of the triangle inequality. The triangle inequality is a relationship between 3 points in X. You need to show that the statement above, which involves n points in X, holds.

For the second part, I attempted a few solns using a typical approach with the triangle inequality, however none of them seemed to be working... Any ideas?