I want to show that if $R$ is reflexive and transitive then $R^{-1}$ is also.
Transitivity: $$(a,b)\in R^{-1} \wedge (b,c)\in R^{-1} \Rightarrow (b,a)\in R \wedge (c,b)\in R \Rightarrow (c,a)\in R \Rightarrow (a,c) \in R^{-1}$$
Reflexivity: $$(a,a)\in R^{-1} \Rightarrow (a,a) \in R$$
Reflexivity seems little bit odd, is it correct or should I add something?
Your proof of reflexivity is perfect.
For transitivity, not quite. We assume that $(a,b),(b,c)\in R^{-1},$ but we don't want to assume that $(a,c)\in R^{-1}.$ We want to conclude that. As a hint for how to do that, since $(a,b),(b,c)\in R^{-1},$ what are some elements that you know must be in $R$?
Edit: So long as you justify each step, it looks good.