Variation of a function and some conclusions

Question

My definition of variation is:
$V_{a}^{b}(f) = \sum_{i=1}^n |f(x_i)-f(x_{i-1})|$ for bounded functions on $[a,b]$
$V_{a}^{b}(f) = \sup \bigl(\sum_{i=1}^n |f(x_i)-f(x_{i-1})|\bigr)$ for functions that might not be bounded on $[a, b]$
Of course $[a, b]$ is decided on $n$ segments and $a = x_0, x_1, ..., x_n = b$
One of the properties of variation of a function says that:
$V_{a}^{b}(f + g) \le V_{a}^{b}(f) + V_{a}^{b}(g)$
Let's define function $h = f - g$
Thus there is a conclusion that:
if functions $f$ and $g$ are nondecreasing then $h \in BV([a, b])$.
Now my question is if the conclusion is true only for functions $f, g$ which are bounded on $[a, b]$?
And another one - is the property true for all functions $f, g$ or again, only those which are bounded on $[a, b]$?