Consider $f, g \in C([0,1])$ There are two different metrics: $d_1(f,g)=\max_{x \in [0,1]}|f(x)-g(x)|$ and $d_2(f,g)=\inf[\epsilon \geq 0; X\subset Y_\epsilon$ AND $Y\subset X_\epsilon ]$ where $X=\operatorname{graph}(f)$ and $Y=\operatorname{graph}(g)$ Note here that $X_\epsilon$ is the set of all points within $\epsilon$ of $X$.
I have encounted a claim that $(C([0,1], d_1)$ is a complete metric space, but that $C(0,1),d_2)$ is not complete. I am failing to see how to see one can be complete but not the other. It seems to me that these metrics are equivalent, so I think feel that either both metric spaces should be complete or both not. Any tips or hints would be much appreciated.
For $n\in\mathbb N$, $n\gt 2$, take:
$$f_n(x)=\begin{cases}0 & 0\le x\le\frac{1}{2}-\frac{1}{n}\\ 1 & \frac{1}{2}+\frac{1}{n}\le x \le 1 \\ \text{linear} & \frac{1}{2}-\frac{1}{n}\le x \le \frac{1}{2}+\frac{1}{n}\end{cases}$$
In other words, $f_n(x)$ has a graph which is a polygonal line $ABCD$ where $A=(0, 0)$, $B=(\frac{1}{2}-\frac{1}{n}, 0)$, $C=(\frac{1}{2}+\frac{1}{n}, 1)$ and $D=(1,1)$
Now this is a Cauchy sequence wrt. $d_2$ that does not converge. (Proof: exercise.)