I have done a course on topology and I am self studying concepts that were not taught in class from the book Foundations of Topology ( C Wayne Patty).
I got struck upon this theorem.
My question is in line 4 of the theorem - How does author wrote $< x_n>$ Union { x} is compact.
Image of Theorem 1.41 ->
I couldn't think how it must hold. Please help.
If $(U_i)$ is an open cover of $<x_n> \cup \{x\}$ then $x \in U_j$ for some $j$. Since $x_n \to x$ there exists $m$ such that $x_n \in U_j$ for all $n \geq m$. For $1\leq n <m$ there exists $U_{i_n}$ such that $x_n \in U_{i_n}$. Now $U_j, U_{i_1},U_{i_2},...,U_{i_{m-1}}$ cover $<x_n> \cup \{x\}$.