Metric space Triangular inequalities

[beetle2]

Homework Statement

let <br /> (X,\sigma) be a metric space. xyz \in Rshow that
<br /> \mid \sigma(x,z)-\sigma(y,z) \mid \leq \sigma(x,y)

Homework Equations

The Attempt at a Solution

\mid \sigma(x,z)-\sigma(y,z) \mid \leq \sigma(x,y)=\mid \sigma(x,y) \mid = \mid\sigma<br /> (z,x) + \sigma(z,y) \mid

\leq \mid\sigma<br /> (x,z)\mid + \mid \sigma(y,z) \mid


Does that look alright?

 

[VeeEight]

For real numbers a and b, |a+b| <br /> <br /> \leq <br /> |a| + |b|.
Thus, |a| + |b|.= |(a-c) + (c-b)| <br /> <br /> \leq <br /> |a-c| + |c-b|