Sure no problem d18ge.
FINAL UPDATE: After doing alxxhrvth's 4 steps a couple of days ago (as I posted about a few replies ago), I still ended up getting the offline indicator for 5 of the 6 gmail accounts that I re-added to Mac Mail. It would happen after Mail.app was hidden for approximately 2 or more hours. So, why not the 6th gmail account? That question I had gave me an idea. Well, if you go to the menu bar in Mail-->Accounts you can adjust 4 checkboxes for each gmail account. They are Mail, Contacts, Calendars and Notes (M, Co, Ca, N). When you setup a gmail account it defaults with all 4 checked. That is probably why alxxhrvth's 4 step program works for the majority. Because it makes us setup an account back to default. But, some of us uncheck the boxes right away. When I was doing alxxhrvth's 4th step the other day, I had left one of the 6 gmail accounts, let's name it Eureka@gmail.com, with the dafault 4 boxes checked. On the other 5, I had unchecked the Co, Ca, N so that only Mail was checked. Yesterday, I checked my inbox and Eureka@gmail didn't have an offline indicator and the other 5 did. So, I messed with the 4 checkboxes on 4 of the 5 troubled gmail accounts. I did different combinations of checkboxes and then I let it run for the last 24 hours. Today, when I went to my inbox after it hiding for more than a couple of hours, they were all online except the one that I left with only mail checked. It didn't seem to matter if I checked M and N, or M and Ca, or M, N, and Ca. Just as long as M (obviously) and one of the other 4 boxes is checked.
CONCLUSION: Make sure that another of the four boxes besides Mail is checked in Mail-->Accounts-->youremail@gmail.com