Intersection of a recognizable language and a decidable language is decidable? I'm having trouble with proving that "Intersection of a recognizable language anda  decidable language is decidable. 

I assume this is true although I have no idea how to proof it. Can somebody point me in the right direction?
It's false. Let $L_1=\Sigma^*$ be a decidable language and $L_2=L_{HALT}$ be the (recognizable) language of all halting TM-string pairs. Then $L_1\cap L_2=L_2$, which is not decidable.