How is this classical group $\textit{compact}$?

Question

Let $O(n)$ be the group of orthogonal $n \times n$ matrices. Apparently this is a "compact classical group" but I have trouble seeing that it is compact. The topology is the topology is inherits from $\mathbb{R}^{n^2}$ but doesn't this mean the space has to be bounded in order for it to be compact? I feel as if there are orthogonal matrices with entries that can be arbitrarily large. Am I wrong?

Book I am using: Homotopical Topology by Fuchs pg 20.

Answer 1

Each entry $a_{ij}$ of an orthogonal matrix $A$ satisfies $|a_{ij}| \le 1$. This is because an $n \times n$ matrix over $\mathbb{R}$ is orthogonal if and only if its columns are orthonormal, so in particular its columns all have norm $1$.

Answer 2

Intuitively, $O(n)$ represents rotation and/or reflection. So, any entry in the matrix may not exceed 1 in absolute value, since otherwise the length in some direction is not preserved.