Let $C[a,b]$ be the space of all continuous function defined on interval $[a,b]$. Consider these two norms and metrics:
$$\|f\|_\infty= \sup_{x\in[a,b]}|f(x)|\text{ and metric }\rho(f,g)=\|f-g\|_\infty$$
$$\|f\|_1=\int_a^b|f(x)|\,dx\text{ and metric }\rho(f,g)=\|f-g\|_1$$
Why the first one is complete while the second is not?
Hints:
1) For the sup norm, take a Cauchy sequence, observe it is pointwise Cauchy, use the completeness of $\mathbb{R}$ to find a pointwise limit, check the limit is uniform. Then it is the well-known fact that the uniform limit of a sequence of continuous functions is continuous.
2) For the $L^1$ norm, consider $f_n(x)$ equal to $0$ on $[a,\frac{a+b}{2}-\frac{1}{n}]$, to $1$ on $[\frac{a+b}{2}+\frac{1}{n},b]$, and connect these two pieces by an affine segment.
