# Metric space Triangular inequalities

1. Feb 10, 2010

### beetle2

1. The problem statement, all variables and given/known data

let $$(X,\sigma)$$ be a metric space. $$xyz \in R$$show that
$$\mid \sigma(x,z)-\sigma(y,z) \mid \leq \sigma(x,y)$$

2. Relevant equations

3. The attempt at a solution

$$\mid \sigma(x,z)-\sigma(y,z) \mid \leq \sigma(x,y)=\mid \sigma(x,y) \mid = \mid\sigma (z,x) + \sigma(z,y) \mid$$

$$\leq \mid\sigma (x,z)\mid + \mid \sigma(y,z) \mid$$

Does that look alright?
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Feb 11, 2010

### VeeEight

For real numbers a and b, |a+b| $$\leq$$ |a| + |b|.
Thus, |a| + |b|.= |(a-c) + (c-b)| $$\leq$$ |a-c| + |c-b|