In "Lie Groups, Lie Algebras and Representations" by Brian C. Hall, a matrix Lie group is defined in sec. 1.1 in the following way: A matrix Lie group $G$ is a subgroup $G \leq \mathrm{GL}(n,\mathbb{C})$ with the following property: If $(A_m)_m^\infty$ is a sequence in $G$ that converges to some matrix $A$, then either $A \in G$ or $A$ is not invertible (i.e. $A \in M_n(\mathbb{C})$ but $A \notin \mathrm{GL}(n,\mathbb{C})$). This is equivalent to saying that $G$ should be a closed subgroup of $\mathrm{GL}(n,\mathbb{C})$. See also the discussion in this Math SE question.
Later, in the same book in sec. 1.3, a compact matrix Lie group is defined as a matrix Lie group $G \leq \mathrm{GL}(n,\mathbb{C})$ which is compact in the relative topology of $M_n(\mathbb{C})\cong \mathbb{R}^{2n^2}$. It is further stated (by the Heine-Borel theorem) that $G$ is compact if and only if i) For any sequence $(A_m)_m^\infty \in G$ that converges to some $A$, then $A \in G$ (i.e. closure) and ii) There exists $C \in \mathbb{R}$ such that $\forall A \in G$ and $\forall j,k \in \{1,...,n\}$ we have $|A_{jk}| \leq C$ (i.e. boundedness).
My question is simply if condition i) above is not redundant, since this closure property is already part of the definition of $G$ as a matrix Lie group. It can be noted that in condition i), it is not stated that $A$ is necessarily in $\mathrm{GL}(n,\mathbb{C})$, i.e. $A$ could be any matrix in $A \in M_n(\mathbb{C})$. Is this the important distinction? If so, what would be an example of a matrix Lie group $G$ that is not closed in the sense of ii)?