Let a, b $\in \mathbb R$ such that a < b. Given $n \in \mathbb N$, consider the following partition of the interval [a, b]:

Question

Let a, b $\in \mathbb R$ such that a < b. Given $n \in \mathbb N$, consider the following partition of the interval [a, b]: $$\qquad P_n = \{ a + j\frac{(b-a)}{2^n}:0 \le j \le 1\} \qquad$$ Sketch the $P_n$ partitions for some values of $n$ and show that. $$\overline \int_a^b f \le \lim \limits_{x\to \infty} S(f;P_n) \qquad$$

My try of the attempt :

$$\overline \int_a^b = inf_p S(f;p) = \sum_{i=0}^n inf_p (t_i - t_{i-1}) = inf_p((t_1-t_0)+(t_2-t_1)+(t_3-t_2)+...+(t_n-t_{n-1}))= inf_p(t_n-t_0) = 0(t_n-t_0)=0 $$

because $ïnf=0$, there is no $sup$.

I thank in advance for any help.