Yes, if you're using a 100 buy-in as your guide, $20,000 is the proper amount to play $1/$2 (100 of 100 big blind buy-ins). Some people use 20 buy-ins, others use 40, 60 or 100. Determine how conservative you want to be and find something that works.
Personally, for online, I am using 60 buy-ins as my bankroll management, but I'll use my personal money to go play $200 at the casino live $1/$2 if I want from time to time, knowing if it's gone, it's all gone (but I have a decent job and could do this a couple times a month if I wanted).
If I run my bankroll up online, I will eventually convert some of that to a live roll and use bankroll management for live games as well.