Proof of Domination Theorem

Question

Prove that if $f(x)$ and $g(x)$ are continuous on $[a, b]$ and $f(x) \leq g(x)$ on $[a, b]$ then $$\int_a^bf(x)\,dx \leq \int_a^bg(x)\,dx$$ Let us divide the interval $[a, b]$ into $n$ equal subintervals of width $\frac{b-a}{n}$ and take the upper sum then we have $$\frac{b-a}{n}\sum_{k=1}^n g\left(a+k\frac{b-a}{n}\right) \geq \frac{b-a}{n}\sum_{k=1}^n f\left(a+k\frac{b-a}{n}\right).$$ Now taking the limits on both sides as $n \to \infty$ we get $$\int_a^bf(x)\,dx \leq \int_a^bg(x)\,dx$$ as the Definite integral is independent from the choice of partition and the choice of $c_k$ $$$$ Is My Proof Correct??