I am sorry for my incompete post, I have updated it.
To prove the finitness of a set one has to establish a bijective correspondence with a finite set, say $z_+$. Let $A$ be a set; let $a_0\in A$. Then there exists a bijective correspondence $f$ of set $A$ with the set $\{1,\ldots,n+1\}$ iff there exists a bijective correspondence $g$ of the set $A-a_0$ with the set $\{1,\ldots,n\}$.
To prove the converse in Topology by James Munkres (2nd edition), Lemma 6.1, the argument made is:
Assume there is a bijective correspondence $f:A\to\{1,\ldots,n+1\}$. If $f$ maps $a_0$ to $n+1$, the restriction $f|A-a_0$ is the desired bijective correspondence of $A-a_0$ with $\{1,\ldots,n\}$.
Otherwise, let $f(a_0)=m$, and let $a_1 \in A$ such that $f(a_1)=n+1$. Then $a_1\neq a_0$. Define a new function $h:A\to\{1,\ldots,n+1\}$ by setting
$h(a_0)=n+1$
$h(a_1)=m$
$h(x)=f(x)$ for $ x \in A-\{a_0,a_1\}$, and here $h$ is bijective. The restriction $h|A-\{a_0\}$ is the desired bijection of $A-\{a_0\}$ with $\{1,\ldots,n\}$.
My question is the if someone has examples of these hypothetical $f,h$ functions, I tried few but not really getting .