$\int_{[a,b]}^{} f(z) dz = \lim_{n\to\infty} \int_{[a_n,b_n]}^{} f(z)dz $

Question

I have two sequences $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ on $\mathbb{C}$ that converge to $a$ and $b$ respectively. I have an open set $U$ on $\mathbb{C}$ which has the segment $[a,b]$ inside, and $f: U \rightarrow \mathbb{C}$ a continous function. I need to prove that $$\int_{[a,b]}^{} f(z) dz = \lim_{n\to\infty} \int_{[a_n,b_n]}^{} f(z)dz .$$ I know that $$\lim_{n\to\infty} \int_{[a_n,b_n]}^{} f(z)dz = \lim_{n\to\infty} (b_n - a_n) \int_{0}^{1} f(a_n + t(b_n - a_n)) dt$$ so i just need to prove that the function sequence $f(a_n + t(b_n - a_n))$ converges to $f(a + t(b - a))$ (uniform convergence), can you please help me proving that that function sequence converges(uniform convergence)? Thank you