Let $A$ a infinite set, and $a$ the cardinal number of $A$, then $a.a=a$

**My attempt:**

We know $a.a=card(A\times A)$ and $card(A)=a$, then we need prove $card(A\times A)=card(A)$

Let $f:A\times A\rightarrow A$ such that $f(x,y)=x$ with $x,y \in A$ this is a bijective function and this implies $card(A\times A)=card(A)$

is correct this?