Non-compactness of $(1,3)$ and $[x,\infty)$ only using the definition

Question

Working from the definition of compactness, determine which of the following subsets of R are compact and prove your assertion.

1. $(1,3)$
2. $[x,\infty)$ where $x$ is finite

I know that both of these are not compact. I know that I need to show that there is a open cover that does not contain a finite sub cover. For 1, The problem is at the points 1 and 3. and for 2 the problem is at infinity. Any help would be appreciated. Thanks

Answer 1

$\bigcup_n (1+\frac{1}{n},3)$ for $(1,3)$ is an open cover with no finite subcover. Similarly $\bigcup_n(x-\frac{1}{n}, n)$ for the second one.