Determining if a sequence of functions is a Cauchy sequence? Show that the space $C([a,b])$ equipped with the $L^1$-norm $||\cdot||_1$ defined by $$ ||f||_1 = \int_a^b|f(x)|dx ,$$
is incomplete. 
I was given a counter example to disprove the statement:
Let $f_n$ be the sequence of functions:
$$f_n(x) = \begin{cases} 0 & x\left[a,\frac{b-a}{2}\right)\\
nx-n\frac{(b-a)}{2} & x\in\left[\frac{b-a}{2},\frac{b-a}{2}+\frac{1}{n}\right)\\
1 & x\in \left[\frac{b-a}{2}+\frac{1}{n},b\right] \end{cases}.$$
This is a cauchy sequence that converges to a discontinuous function. 
My question is:
How do I see that such a sequence of functions is cauchy? My thought was that the $||\cdot||_1$ will determine the differences in area under the curve for each function, so that $||f_n-f_m||\leq \frac{(b-a)}{2}$. Is this correct?

I don't think you need to be so precise in finding a counterexample, you just have to show the existence of one. First, recall that the step functions are dense in $\mathcal{L}^1([a,b])$. Then, if you slope the sides of the characteristic funtions somewhat (this can be made precise, see any book on real analysis), you get a set of continuous functions dense in $\mathcal{L}^1([a,b])$. But there are most definitely non continuous functions in this set.
You could get to the same thing using Stone-Weirstrass also, but that is even less concrete. 
