Let $\{f_n\}_{n=1}^{ +\infty} $ be a Cauchy sequence for $\lVert\cdot\rVert$. In particular, the sequence of real numbers $\{f_n(0)\}$ is Cauchy, hence converges to a real number we call $f(0)$. Now, considering the partition $t_0=0<1=t_1$, we have
$$\lVert f_k-f_j\rVert\geqslant\operatorname{Var}(f_k-f_j)\geqslant \left|f_k(1)-f_j(1)\right|-\left|f_k(0)-f_j(0)\right|,$$
proving that $\left\{f_k(1)\right\}$ is Cauchy, hence converges to a real number called $f(1)$. Now for $t\in(0,1)$, we consider the partition $t_0:=0
$f$ is of bounded variation. Indeed, let $t_0=0
$\lVert f-f_N\rVert\to 0$. We have by definition $f_n(0)\to f(0)$ so we have to show that $\operatorname{Var}(f_n-f)\to 0$. Let $\varepsilon>0$. We can find $N=N(\varepsilon)$ such that if $m,n\geqslant N$ and $0=t_0