Even numbers = $\{ 2,4,6,8\}$
Odd numbers = $\{ 1,3,5,7,9 \}$
Sum property =
Choosing $2$ numbers possibility = $\{EE,\color{Red}{EO},\color{Red}{OE},OO\}$
Example $3+2=5$ Fail
Choosing $3$ number possibility = Use pigenhole no matter how you choose $3^{rd}$ number either will be $odd$ or $even$.
$\therefore$ Sum will be even.
Example $ {\{3,2\}}$ Choice $Odd$ or $Even $ lets say $5$ or $4$
Then , $5+3=8$ or $4+2=6$