a. Show that every infinite set can be put into a bijection with a proper subset of itself.
b. Show that the initial segment determined by $n$ cannot be put into a bijection with the initial segment determined by $m\in\mathbb{N}$, if $m<n$.
c. Show that $\mathbb{N}$ cannot be put in a bijection with any initial segment of $\mathbb{N}$.
a. : I know that every infinite set has a denumberable subset, but I don't understand how it can have a bijection to the subset since its infinite?
b. and c. : Do not know how to do. Would it be true the other way around?
Hints:
a. If $A$ is an infinite set, $A'=\{a_n\mid n\in\mathbb N\}$ is a denumerable subset, then $f(a)=a$ for $a\notin A'$, and $f(a)=a_{n+1}$ whenever $a=a_n$ for some $n$ is a bijection of $A$ with a subset of itself.
b. Suppose there was an injection from $\{0,\ldots,n\}$ to $\{0,\ldots,m\}$, and $m<n$, then we can reiterate the injection because the elements in the range are also in the domain. Show that every iteration must omit some elements from the range, and so after finitely many steps we must have a function from a non-empty set into an empty set which is absurd.
c. $\mathbb N$ can be put in bijection with some of its proper subsets. Show that this property is preserved under bijections, and deduce from b. that $\mathbb N$ cannot be injected into any of its proper initial segments.