Understanding uniformly bounded sequence The sequence $(a_i)_{i\in I}$ is uniformly bounded if there exists a number $M<\infty$ such that $|a_i|\leq M,\forall i\in I$. 
Why can't I say that the sequence is uniformly bounded when $|a_i| <\infty,\forall i\in I$?

Well, take $a_i:=i$. Clearly each $a_i$ is bounded, but there is no common bound for all of the $a_i$, since $a_i\to\infty$ as $i\to\infty$.
Another way of seeing that the sequence is not uniformly bounded is to plot the $a_i$ : the graph doesn't lie in a horizontal strip.
