Every Cauchy sequence in $\{f\in (C([0,1]),\|\cdot\|_1)\,|\,\exists a,b\in\mathbb R:f(x)=ax+b\}$ converges

Question

I have trouble proving that, using the norm $\|f\|=\int_0^1|f(x)|\mathrm dx$, for a Cauchy sequence of functions $f_n(x)=a_nx+b_n$, the sequences $(a_n)_n$ and $(b_n)_n$ also have to be Cauchy sequences. The only thing I could conclude so far is that for any $\varepsilon>0$ there exists an $N\in\mathbb N$, such that $$\varepsilon>\left|\frac{|a_n-a_m|}2-|b_n-b_m|\right|\ \forall\ n,m>N$$ but I don't see how I'd prove that $(a_n)_n$ and $(b_n)_n$ are Cauchy sequences.

Answer 1

If $(f_n)$ is Cauchy in that norm then you can show that the two sequences $(\int_0^1 f_n)$ and $(\int_0^1 tf(t)\,dt)$ are Cauchy...