I want to show that if $a\lt b$, then $(a,b)$ is not of measure zero.
My idea was to show that any interval covering $(a,b)$ is such that the sum of the lengths of the intervals is always greater than $b-a$.
So I tried induction. Suppose $n=1$, where $n$ is the number of intervals covering $(a,b)$. Then the interval consists of a single set of the form, say, $(c,d)$ such that $(a,b)\subset (c,d)$. If this is the case then obviously, we would have $d-c \gt b-a$, where I have assumed that $c\lt a$ and $b \lt d$.
Now my questions: I am a little bit lost as to how to show it for $n+1$.
Am I even approaching it correctly?
Thanks for your help.