Triangle Inequality Property for the Euclidean Metric

Question

I've read in many of my books that the triangle inequality for a metric space of the Euclidean Metric is defined as:

$$d(x,y) \leq d(x,z) + d(z,y)$$

But when I look up the proof, to help me understand it, many if not all the proofs go about it this way:

$$d(x,y)+d(y,z) \geq d(x,z)$$

I do not understand why they go about it this way, instead of by the definition. What am I missing?

I am using this definition of Euclidean Metric http://mathworld.wolfram.com/EuclideanMetric.html

Answer 1

It is same in the substance, although they are different in the form. The following picture may be helpful for you to understand the triangle inequality.