How do I show that O(n) is compact? [duplicate]

Show that the set of all orthogonal matrices in the set of all $n \times n$ matrices endowed with any norm topology is compact.

Recall a compact subset of $R^{n \times n}$ is a set that is closed and bounded. One way to show closedness is to observe that the orthogonal matrices are the inverse image of the element $I$ under the continuous map $M \rightarrow MM^T$. Boundedness follows for example from the fact that each column or row is a vector of magnitude $1$.

first derivative test to find where the function is increasing, and decreasing

Do the rings $\mathbb{Z}[x]$ or $\mathbb{Q}[x]$ have a quotient isomorphic to the field with 9 elements?

Is my approach to showing $(\exists y\in B\cap C)\Rightarrow(x\in A)$ correct?

Reference for an unknotting move

Show that max $|f(x)| ≥ (c_0/8) (b − a)^2$

Nuclearity of finite dimensional $C^*$-algebras

Hughes and Cresswell's definition of consistency

The dual semigroup is equivalent in norm to its original semigroup

Let $\{A_i\}_{i \in I}$ be a locally finite collection of closed subsets of $X$. Show that the union $\bigcup_{i \in I} A_i$ is closed in $X$.

Suppose that $c$ is transcendental over $\mathbb{Q}.$ Show that $\sqrt{c}$ and $c + \sqrt{c}$ are also transcendental.

Can I store items in my apartment?

What is a good beginners strategy to learn the mechanics of Civilization 5?