## How to prove re-hard or co-re hard by reduction?

My question asks to show that $$FINITE_{TM}$$ is r.e. hard and co.re hard by reduction. My first idea is that we can show $$HALT_{TM}$$ that’s reducible to it. So, my assumption is to prove $$FINITE_{TM}$$ is r.e. hard, we can reduce $$HALT_{TM}$$ to $$FINITE_{TM}$$. And to show $$FINITE_{TM}$$ is co-r.e. hard, we can reduce $$\overline{HALT}_{TM}$$ to $$FINITE_{TM}$$. Does my assumption sound accurate? Any suggestions?