Suppose $f$ is continuous and $φ$ is of bounded variation on $[a, b]$. Show the function $ψ(x) = \int_x^a f dφ$ is of bounded variation on $[a,b]$
I'm unsure how to finish this proof, can someone show me or guide me?
So I know since $\phi$ is of bounded variation, there exists some constant $M$ such that for any partition $\Gamma = \{a=x_0,...,x_m=b\}$, $S_{\Gamma}[\phi,a,b] = \sum_{i=1}^m |\phi(x_i)-\phi(x_{i-1})| \leq V[\phi,a,b] = sup_{\Gamma}S_{\Gamma} \leq M$.
Now I also know that f is continuous on $[a,b]$ thus it is also bounded on $[a,b]$, but I'm not sure how to mix the two together. I had the thought of taking the definition of $V[\psi,a,b]$ and trying to pop out $S_{\Gamma}$ to show the integral is bounded then if the integral is bounded then $\psi$ is of bounded variation, but I couldnt get that to work.
Any help is greatly appreciated.
Edit: Current attempt
Let $\Gamma$ be some partition of $[a,b]$. Let $f$ be continous on $[a,b]$ thus $f\leq M$.
$V[\psi,a,b] = sup_{\Gamma}\sum_{i=1}^{m} |\psi(x_i)-\psi(x_{i-1})| = sup_{\Gamma}\sum_{i=1}^{m} | \int_a^{x_i} fd\phi - \int_a^{x_{i-1}} fd\phi | = sup_{\Gamma}\sum_{i=1}^{m} | \int_{x_i}^{x_{i-1}}f d\phi | \leq sup_{\Gamma}\sum_{i=1}^{m} |M(\phi(x_i)-\phi(x_{i-1}))| = M* sup_{\Gamma}\sum_{i=1}^{m} |\phi(x_i)-\phi(x_{i-1})| = M*V[\phi,a,b] < \infty$ thus $\psi$ is of bounded variation.
Is this correct?
Let $x \mapsto V_a^x$ denote the total variation function for $\phi$, i.e., $V_a^x$ equals the total variation of $\phi$ over the interval $[a,x]$.
As proved here, we have
$$\left|\int_{x_{i-1}}^{x_i}f \, d\phi\right| \leqslant \int_{x_{i-1}}^{x_i}|f|\,dV_a^x$$
Since $|f| \leqslant M $ and $V_a^x$ is monotone increasing with respect to $x$, it follows that
$$\left|\int_{x_{i-1}}^{x_i}f \, d\phi\right|\leqslant M \int_{x_{i-1}}^{x_i}\,dV_a^x = M(V_a^{x_i} - V_a^{x_{i-1}})= MV(\phi,x_{i-1},x_i)$$
To finish your proof, note that
$$\sum_{i=1}^n |\psi(x_i) - \psi(x_{i-1})| = \sum_{i=1}^n\left|\int_{x_{i-1}}^{x_i}f \, d\phi\right| \leqslant M \sum_{i=1}^nV(\phi,x_{i-1},x_i) = MV(\phi,a,b)$$