The sequence $\{S(f,P^t_n)\}$ converges to $\int_a^b f(x)dx$ ( Proof with the traditional definition )

Question

Theorem :

Assume that $f:[a,b] \to \mathbb R$ is Riemann-integrable and $\{P_n^t\}$ is a sequence of labeled partitionings such that $\{||P_n^t||\}$ converges to $0$. Then the sequence $\{S(f,P^t_n)\}$ converges to $\int_a^b f(x)dx$.

My idea to prove this :

First we show that $\{S(f,P^t_n)\}$ is Cauchy . We conclude that its convergent and then we use the definition of Riemann-integration. The problem is i can't write my idea... I'm stuck at proving $\{S(f,P^t_n)\}$ is Cauchy.

Note 1 : My definition of integration is the traditional definition which Riemann introduced.

Note 2 : $S(f,P^t_n)$ means the Riemann summation of $f$ with respect to the labeled partitioning $P^t_n$.

Thanks in advance.