My mother is in the later stages of Alzheimer's. She doesn't talk anymore, just babbles a little, and my brother insisted a couple of years ago that she have a feeding tube put in because she forgets to chew and swallow. I was against it. I felt it was more honorable for mom to enter hospice and go naturally, but he blew up at me and accused me of "killing Mom." Now, he'll barely talk to me and will hardly let me see her.
My mother's life is so sad. She can't do anything but sit or lay down, and she tries constantly to put anything she can fit into her mouth. How do I convince him that it's time to let Mom go?
It's very hard on families to come to terms with end-of-life decisions. As always, I encourage you to have a heart-to-heart talk. Assure your brother that you love your mom and, of course, wish she could live a long, healthy life—but is this living? Really listen to his responses. Is fear driving him? Is he afraid he'll feel responsible for "killing" his mom? Even if he resists, try several times to discuss it with him. Talk, cry, ask questions, yell if you have to—don't give up easily.
If he still won't tell you, can you privately guess at his motives? Could it have anything to do with feelings from childhood? As much as we like to think that we've moved beyond our childhood, it's still there, and often we try to "right" the past by controlling the present. This is scary for him, as I'm sure it is for you. His stubbornness is probably covering up guilt, anxiety, and grief.
Also take a step back, and try not to view this situation as your brother is wrong and you're right. Many people would agree with him—that we should do all we can to keep someone alive. Others would concur with your viewpoint and choose to let a family member die naturally with as little intervention as possible. Your goal is to come to an agreement with your brother to do what's best for your mom.
Explain to your brother that hospice care honors the inevitability that we're all going to die and treats the end-of-life passage with dignity. Go to hospice and get their literature on dying, and ask the hospice staff to share their philosophy with your brother.
If, after all your efforts to communicate with him, your brother is still shutting you out, find out the legal situation. If your brother has your mother's durable power of attorney, that means he has the responsibility to manage her finances, not necessarily her health. You shouldn't have been pushed out of your mom's life. It's time that you step back in and demand to be treated as an equal sibling. What have you got to lose? If he's already barely speaking to you, then don't worry about "ruffling his feathers." You have every right to visit your mother. Consider visiting when you know he's not there so that you can have a few quiet moments with her.
You might also have to accept that barring legal intervention, you can do little to change this situation. It's not an uncommon one, and in the end, it's best to not turn this into a family feud. Value whatever amount of time you have with your mom and make sure she's getting good care. Continue to reach out to your brother and love your mom. She may not be able to acknowledge that she knows you, but love is action and your diligence is the best way to express it.