How do I convince someone it's best to put a dog out of their misery?

At least I think it's best. Let me explain. My sister has a beautiful, kind, gentle German Shepard/Pitt Bull mix who is now very elderly and in bad condition. Once he was the picture of physical health but now he's brittle with arthritis to the point he strains to move, can no longer run, has seizures... show more At least I think it's best. Let me explain.

My sister has a beautiful, kind, gentle German Shepard/Pitt Bull mix who is now very elderly and in bad condition. Once he was the picture of physical health but now he's brittle with arthritis to the point he strains to move, can no longer run, has seizures often and just seems like an unhappy dog. Her vet has thrown hints around about putting him to sleep but either my sister didn't catch them or is just refusing to think about it. This dog had a fantastic life and is about 17 years old now but I feel like it's time she let him go for his own good. I hate to see him this way.

I'm afraid to mention it to her though because in the past when I did she got tearful and told me I was heartless which hurt me. I just want her to see that sometimes the best thing isn't what you want to hear...
Update: I'm 21 and my sister is 17. She's had this dog since she was a few months old.
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