Prove or disprove that if a relation $R^2$ is transitive then $R$ is also transitive

Question

Prove or disprove that if $R^2$ is transitive then $R$ is also transitive.

I tried to prove $(R\circ R)^2\subseteq (R\circ R)\implies R^2\subseteq R$

this way

$(R\circ R)\circ (R\circ R)\subseteq (R\circ R)\implies R^2\subseteq R$

$R\circ (R\circ R) \circ R\subseteq (R\circ R)\implies R^2\subseteq R$

$R\circ R\subseteq (R\circ R)\implies R^2\subseteq R$

I saw this "pattern" somewhere so I tried to use it but it doesnt seem to be right way

Answer 1

$R:=\{(1,2),(2,1)\}$ is not transitive. $R\circ R=\{(1,1),(2,2)\}$ is transitive.