On page 12 of Luenberger's Optimization by Vector Space Methods, he claims that the set of bounded real sequences is vector space with addition defined component-wise.
Ok, but what is the underlying field? $\mathbb{R}$? But then for a given $K$, you can always find a large enough $M(K)\in \mathbb{R}$ such that $M(K)\left(x_i\right)_{i\in \mathbb{N}}$ becomes unbounded ($\exists i\in\mathbb{N}$ such that \lvert M(K)x_i\rvert > K) and so is not closed under scalar multiplication.
Thinking naively, if it is not possible to pick an arbitrarily large scalar, then it should also not be possible to ever construct an unbounded real sequence (as the components of the sequence and the field are the same set, $\mathbb{R}$).
As is probably evident, my knowledge of analysis is rudimentary.