# Defintion (s) > [!NOTE] Definition (Bounded Real Sequence) > We say that a [[Real sequences|sequence]] $(a_{n})$ is bounded iff $\exists A \in \mathbb{R}, \, A>0, \quad \text{s.t. } \quad |a_{n}|\leq A \quad \text{for every } n \in \mathbb{N}$so the set is bounded above by $A$ and bounded below by $-A$. **Negation** gives that $(a_{n})$ is unbounded if $\forall A \in \mathbb{R},\, A>0, \quad |a_{n}| > A \quad \text{for some }n \in \mathbb{N}$and we write $a_{n} \to \infty$ as $n \to \infty$. > [!NOTE] Definition (Bounded Sequence in $\mathbb{R}^n$) > Let $(x_{j})$ be a sequence in $\mathbb{R}^n$. We say that $x_{j}$ is bounded is bounded there exists $M>0$ such that for all $j\in \mathbb{N}$, $\lVert x_{j} \rVert \leq M$ where $\lVert \cdot \rVert$ denotes the [[Euclidean Norm]].