Which of these are Cauchy Sequences?

Question

Which of these are Cauchy Sequences?

A. $f_n(x):=0$ if $x\notin [n-1,n+1]; x-n+1$ if $x\in[n-1,n]$,$n+1-x$ if $x\in[n,n+1]$

and $X=\{f:\mathbb R \to \mathbb R| f iscontinuous \int_{-\infty}^{\infty}|f(t)|dt <\infty\}$ with $d_1$ metric,

The $d_1$-metric on a space of functions defined over a domain $X⊂\mathbb R$, whenever it is well-defined, is defined as follows:

$d_1(f,g):=\int_X|f(x)-g(x)|dx.$

B.$f_n(x)=\frac{x+n}{n}$ in $C[0,1]$ with usual sup-norm metric

C.$f_n(x)=\frac{nx}{1+nx}$ in $C[0,1]$ with usual sup-norm metric

Answer 1

In B you should use the sup-norm metric !

Hence you have to investigate $d(f_n,f_m) =sup \{|f_n(x)-f_m(x)| : x \in [0,1]\}$

Show that

$|f_n(x)-f_m(x)|=|\frac{1}{n}-\frac{1}{m}|x$

for all $x \in [0,1]$.