Let $\{f_n\}_{n \in \mathbb{N} }$ be a sequence of functions $f_n:I\rightarrow \mathbb{R}^d$ where $I \subset \mathbb{R}$ interval.
A sequence of function is not uniformly bounded if $\forall K > 0$ there exist $n \in \mathbb{Z}^+, t\in I$ such that $\|f_n(t)\| > K$.
I'm looking for a characterization in terms of sequences.
I can see that if $f_n$ is bounded for each $n$ then these two are equivalent:
1.there exists $t_n\in I$ with $\lim_n\|f_n(t_n)\| \rightarrow +\infty$
2.the sequence is not uniformly bounded.