Prove that a set $X$ is infinite if and only if $X$ is equivalent to a proper subset of itself.
If $X$ is finite, then suppose $|X|=n$. Any proper subset $Y$ of $X$ has size $m<n$, and so there cannot be any bijective mapping between $Y$ and $X$.
If $X$ is countably infinite, then suppose $X=\{x_1,x_2,\ldots\}$. We can map $X$ to $Y=\{x_2,x_3,\ldots\}$ by using the map $f(x_i)=x_{i+1}$.
But what if $X$ is uncountably infinite? How can we specify the mapping?
Best Answer
The solution was in the previous title: if $X$ is infinite, then it contains a countable infinite subset, say $X_0$. Then you gave a bijection $X_0\to X_0\setminus\{x_1\}$, that extends to a bijection $X\to X\setminus\{x_1\}$, that acts as identity on $X\setminus X_0$.