True or False?

a)The $\sup \{\frac{(n-1)}{n} : n\in\mathbb{B} \}$ exists?

b) If ${x\_n}$ is a sequence with $x\_n < 3$ for all $n$, then it cannot converge to $3$?

For a) True: By definition, the number $s$ is called the supremum of a set $E$ if 1) $S$ is an upper bound for $E$. And 2) If $t$ is an upper bound for $E$, then $s \leq t$.

Then take $\sup \{1 - \frac{1}{n} : n \in \mathbb{N} \}$.
Then we can see that $1$ is an upper bound for the set. And if $t$ is an upper bound for the set, then $1 \leq$ for all $t$. 

For b) True : Take $x\_n = (-1)^(n)$ , then this is bounded above by $1$ and bounded below by $-1$, then $x\_n < 3$. Then it cannot converge to $3$.

Can someone please verify this?

Thank you.