$A,Bsubsetmathbb{R}, |A|<inftyRightarrow |B-A|geq |B|-|A|$

I have proved the following statement and I would like to know if I have made any mistakes, thanks.

"$A,Bsubsetmathbb{R}, |A|<inftyRightarrow |B-A|geq |B|-|A|$"

where $|cdot|$ denotes outer measure.

My proof:

(1) countable subadditivity of outer measure; (2) outer measure preserves order

(I) $|B|=|B-Acup Acap B|overset{(1)}{leq}|B-A|+|Acap B|Rightarrow |B-A|geq |B|-|Acap B|$

(II) $Acap Bsubset Aoverset{(2)}{Rightarrow} |Acap B|leq |A|Rightarrow -|Acap B|geq -|A|Rightarrow |B|-|Acap B|geq |B|-|A|overset{(I)}{Rightarrow} fbox{$|B-A|geq |B|-|A|$}$