I wonder why Michael Spivak assumed $f$ is integrable on $[a,b]$ "Calculus 4th Edition" by Michael Spivak "Measure, Integration & Real Analysis" Axler

Question

The following theorem is in "Calculus 4th Edition" by Michael Spivak.

THEOREM 1 (p.503)
Suppose that $\{f_n\}$ is a sequence of functions which are integrable on $[a,b]$, and that $\{f_n\}$ converges uniformly on $[a,b]$ to a function $f$ which is integrable on $[a,b]$. Then $$\int_a^b f=\lim_{n\to\infty} \int_a^b f_n.$$

The following is Exercise 14 in Exercises 1A in "Measure, Integration & Real Analysis" by Sheldon Axler.

Suppose $f_1,f_2,\dots$ is a sequence of Riemann integrable functions on $[a,b]$ such that $f_1,f_2,\dots$ converges uniformly on $[a,b]$ to a function $f:[a,b]\to\mathbb{R}$. Prove that $f$ is Riemann integrable and $$\int_a^b f=\lim_{n\to\infty} \int_a^b f_n.$$

I wonder why Michael Spivak assumed $f$ is integrable on $[a,b]$.

Any reason?

Answer 1

I found the following Problem (Problem 26 on p.523 in "Calculus 4th Edition" by Michael Spivak).

26 (a)
Prove that if $f$ is the uniform limit of $\{f_n\}$ on $[a,b]$ and each $f_n$ is integrable on $[a,b]$, then so is $f$. (So one of the hypotheses in Theorem 1 was unnecessary.)

I also found that Michael Spivak really assumes that the reader will solve all the problems.