I am fairly new to real analysis and shyly looked into the book specified in the title because I loved Linear Algebra Done Right also by Axler.
Now I think Heine-Borel makes $[0,1]$ compact and a continuous real-valued function on a compact set is bounded and uniformly continuous. Thus if $f$ is continuous on $[0,1]$, uniform continuity can be used to prove that $f$ is Riemann-Integrable.
Hoping I got the above right, I wondered how to go about Exercise 5 of Section 1B, Axler.
It asks one to give an example of a sequence of continuous real-valued functions $f_1, f_2,…$ on $[0, 1]$ and a continuous real-valued function $f$ on $[0, 1]$ such that $$ \lim_{k\to \infty} f_k(x) = f(x) $$ for each $x\in [0,1]$. But $$ \lim_{k\to \infty} \int_0^1 f_k \neq \int_0^1 f $$
Just one page above I also found this useful Statement (1.18) about interchanging Riemann integral and limit, Axler.
It tells us that if a sequence of Riemann-Integrable bounded functions $f_1, f_2,…$ on $[a,b]$ has limit $f$ and $f$ is known to be Riemann-Integrable, then $$ (1) \space \lim_{k\to \infty} \int_a^b f_k = \int_a^b f$$.
Now as established in the beginning, we know that the $f_k$’s as well as the limit $f$ in exercise 5 are Riemann-Integrable and bounded (because continuous on compact set). Which means (1) should hold by statement 1.18. But this contradicts the goal of the exercise. So how can one give an example as required?
What am I not seeing i.e. where is my mistake in this? Surely I misunderstood one of the definitions or theorems.
Thank you to anyone reading this.