I am reading "Introduction to Analysis I" (in Japanese) by Mitsuo Sugiura.
The author wrote the following definition on p.302 in this book:
Definition 1
Let $B(A,\mathbb{R}^m)$ be the set of the bounded functions from $A$ to $\mathbb{R}^m$.
For each $f\in B(A,\mathbb{R}^m)$, $$||f||=\sup_{x\in A} |f(x)|$$ is called the uniform norm of $f$.
Let $(f_n)_{n\in\mathbb{N}}$ be a sequence of functions on $A$.
When $\lim_{n\to\infty} ||f-f_n||=0$, we say $(f_n)_{n\in\mathbb{N}}$ converges uniformly to $f$ on $A$.
I wonder why the author didn't assume that $f,f_1,f_2,\dots$ are elements of $B(A,\mathbb{R}^m).$
Any reason?