Q:

Finite quadratic variation leads to finite covariation

I'm trying to prove that if two functions have finite quadratic variation then their covariation is finite. I've seen that $2|[X,Y]_{t}| \leq [X]_{t}+[Y]_{t}$ but I can't see how to get there. It seems like I want to prove $[X,Y]_{t} \leq min\{[X]_{t},[Y]_{t}\}$. 
I was thinking maybe:
$$\sum_{0}^{n}(X_{k}-X_{k-1})(Y_{k}-Y_{k-1}) \leq min\{max\{X_{k}-X_{k-1}\}\sum_{0}^{n}(Y_{k}-Y_{k-1}), \\ max\{X_{k}-X_{k-1}\}\sum_{0}^{n}(Y_{k}-Y_{k-1})\} $$
But I couldn't really justify the argument to myself. Can anyone help?

A:

Note that
$$[X-Y,X-Y] \geq 0$$
and since
$$[X-Y,X-Y] = [X,X]- 2 [X,Y] + [Y,Y]$$
this implies
$$2[X,Y] \leq [X,X] + [Y,Y].$$
Replacing $X$ by $-X$ and using $[-X,-X] = [X,X]$, we get
$$-2[X,Y] \leq [X,X] + [Y,Y].$$
Consequently,
$$2\big|[X,Y] \big| \leq [X,X]+[Y,Y].$$

