I am reading "Understanding Analysis 2nd Edition" by Stephen Abbott.
Define $\lim_{m,n\to\infty} a_{mn}=a$ to mean that for all $\epsilon>0$ there exists an $N\in\mathbb{N}$ such that if both $m,n\geq N,$ then $|a_{mn}-a|<\epsilon.$
The iterated limits are $\lim_{n\to\infty}(\lim_{m\to\infty}a_{mn})$ and $\lim_{m\to\infty}(\lim_{n\to\infty}a_{mn})$.
The following exercises are Exercise 2.3.13 (d) and (e) on p.56 in this book:
(d) Assume $\lim_{m,n\to\infty}a_{mn}=a,$ and assume that for each fixed $m\in\mathbb{N},$ $\lim_{n\to\infty} a_{mn}=b_m.$ Show $\lim_{m\to\infty} b_m=a.$
(e) Prove that if $\lim_{m,n\to\infty} a_{mn}$ exists and the iterated limits both exist, then all three limits must be equal.
I wonder why the author didn't write as follows:
(e)' Prove that if $\lim_{m,n\to\infty} a_{mn}$ exists and for each fixed $m\in\mathbb{N},$ $\lim_{n\to\infty} a_{mn}$ exists and for each fixed $n\in\mathbb{N},$ $\lim_{m\to\infty} a_{mn}$ exists, then $\lim_{m\to\infty}(\lim_{n\to\infty}a_{mn})$ and $\lim_{n\to\infty}(\lim_{m\to\infty}a_{mn})$ exist and $\lim_{m,n\to\infty}a_{mn}$ and $\lim_{m\to\infty}(\lim_{n\to\infty}a_{mn})$ and $\lim_{n\to\infty}(\lim_{m\to\infty}a_{mn})$ must be equal.
I think (e)' is more natural than (e).
Why did the author write (e) instead of (e)'?