Euclidean metric is a metric

I am trying to prove that the euclidean metric is a metric. I was able to prove the first three properties easily but the fourth is giving me trouble.

I have to prove that

Now in the outlined proof they ask you to first prove the CBS inequality and obtain the form

And I've done this.

What I don't get is the jump from this to the proof of the fourth condition.

Could someone give me a hint (I'd actually prefer a hint rather then the whole solution if possible or a guide I want to get as much of this on my own as possible.)

Also, do I need to use and if so how should I approach proving that?