I am helping my father with Alzheimers on a really old Mac (OS X 10.4.11). What alternate solutions
First off, I want to say "thank you" to TeamViewer for providing this great software. It has helped me support my father on his Mac for well over a decade! In the last year it's become crucial because his Alzheimer's progressively robs him of the ability to work with his computer.
It is with sadness but also understanding that I receive the news that TeamViewer will stop supporting his operating system (Mac OS X 10.4) as of June 18, 2019.
"please take a couple of minutes to help your remote partner to update their TeamViewer and enjoy the latest features."
I wish I could. I would've updated a long time ago but I cannot. My father's disease is a prison that forces him to use software first written in 1992. Microsoft Word 5 and SuperPaint 3.2 fortunately still run on a PowerPC from 2004 under Mac OS X 10.4.11 and fortunately TeamViewer has supported Mac OS X 10.4.11 for all these years.
I think I may have a work-around but it will be nowhere near as simple as TeamViewer 
.
Mac OS X has a built in VNC server.
My alternate solutions:
#1 Run a script on my father's Mac that updates a URL to point to my father's computer. Once upon a time I used a free URL service with a homebrew Linux server that had dynamic IP service. The IP address the free URL pointed to would get updated every 15 minutes and I would redirect a URL that I owned to point to the free URL.
Then I can access my father's computer using VNC and the URL.
Problem: VNC is not secure and very easily compromised. Mac OS X 10.4.11 is also older and no doubt has numerous security holes that can be accessed with local access. It also means opening up the computer to the internet. Granted, my father has no financial or personal information on his computer and he has no access to his finances anyway, but, it does provide a pathway to get into my parents' LAN.
#2 Use VNC via an SSH tunnel with solution #1. That would mitigate the risk of compromising the computer since SSH is a robust protocol and has been around for a long time (I guess I'll have to check the SSH version on Mac OS X 10.4.11). Certainly not a solution for the faint of heart.
#3 Run Ubuntu Linux on a mini PC and run TeamViewer on top of that on his network. Access the Ubuntu desktop and then connect to my father's Mac locally via VNC.
Problem: The difficulty with this is that their connection to the internet is SLOW and it introduces yet another moving target (another computer) in what is already an imperfect setup. It's hard enough keeping the Mac connected to the router. Having another PC to troubleshoot will be yet another point for failure.
