The following is a problem I encountered in my textbook on real analysis:
Let us define:
$$ \forall f,g \in C^0([0,1]): d(f,g) = \int_{0}^{1} \frac{|f(x)-g(x)|}{1+|f(x)-g(x)|}dx $$
We are asked to show $ d $ is a metric and to show that it is not complete.
I have managed to show $d$ is a metric but I am stuck on the incompleteness. Could anyone please guide me om showing it is incomplete? Thanks to all helpers.
Let $f_n(x)=0$ for $x \leq \frac 1 2$, $1$ for $\frac 1 2 +\frac 1 n \leq x \leq 1$ and let it be linear between $\frac 1 2$ and $\frac 1 2+\frac 1 n$. Let $f(x)=0$ for $x <\frac 1 2$, $f(x)=1$ for $x \geq \frac 1 2$. Then DCT shows that $\int_0^{1} \frac {|f_n-f|} {1+|f_n-f|} \to 0$. It follows that $(f_n)$ is a Cauchy sequence w.r.t the metric $d$. Suppose there exists $g \in C^{0}[0,1]$ such that $d(f_n, g) \to 0$. Then $f_n \to g$ in measure and hence there is a subsequence which converges to $g$ almost everywhere. This implies that $f=g$ almost everywhere. By continuity $g$ must vanish identically in $(0,\frac 12) $ and $g(x)$ must be $1$ on $(\frac 1 2,1)$. This is a contradiction.
