If $f$ is continuous on $[a,b]$, show that $$\lim_{c\to a^+}\int_{c}^{b}f(x)dx=\int_{a}^{b}f(x)dx$$
Hint: A continuous function on a closed finite interval is bounded and there exists a constant $K>0$ s.t. $\forall x\in [a,b],|f(x)|\le K$.
I want to use the hint above along with the triangle inequality for integrals and the fact that the sum of two integrals with the same integrand and with bounds a,b and b,c is equal to the integral with bounds a,c.
I started with this but quickly got to a point that looks useless: $$|{\int_{a}^{b}f(x)dx}|\le \int_{a}^{b}|f(x)|dx\le \int_{a}^{b}Kdx=K(b-a)$$ Also, it looks like a situation that needs an epsilon delta proof but I've worked on it for a while with little results... Thank you!
HINT $$0 \le \left| \int_a^b f - \int_c^b f \right| = \left| \int_a^c f \right| \le \int_a^c |f| \le \int_a^c K = K|a-c| \to 0$$