Understanding a theorem of uniformly continuous

Question

$\textbf{Theorem:}$ Suppose $I \subset \mathbb{R}$ be a closed and bounded interval. If $f:I\rightarrow \mathbb{R}$ is continuous on $I$, then $f$ is uniformly continuous on $I$.

My question is: why $I$ should be bounded?

Answer 1

If $I$ were not bounded, then a continuous function on $I$ may not be uniformly continuous on $I$. For example, let $I=[a,\infty)$ and $f(x)=x^2$ for $x \in I$. Then $f$ is continuous on $I$ but not uniform continuous. To verify this, take $x_n=a+n$ and $y_n=a+n-\frac{1}{n}$ for $n \in \mathbb{N}$. Then $x_n-y_n$ tends to $0$ whereas $f(x_n)-f(y_n)$ is not.