uniform convergence on bounded sets vs un bounded sets

Question

For the function $f_k(x) = \chi_{[-k,k]}$ then we have that $f_k$ increases to the constant function $f(x) = 1 \forall x$. I am confused by the statement:

convergence is uniform on any bounded set and fails to be uniform on any unbounded set. What exactly does this mean?

I understand that the bounded set in this case is the sequence of sets $[-k,k]$ and the unbounded sets are $\mathbb{R} -[k,k]$, but i don't understand thinking about convergence on different parts of the domain

Answer 1

"convergence is uniform on any bounded set and fails to be uniform on any unbounded set".

It means the following: if $X$ and $Y$ are arbitrary subsets of $\mathbb R$, with $X$ bounded and $Y$ unbounded, then $(f_k|_X)_k$ converges uniformly to $f|_X$ and $(f_k|_Y)_k$ does not converge uniformly to $f|_Y$.