Also, I'd be interested to know how people have convinced their parents that such a thing would be helpful to them!
In my experience you will not convince your mum. I guess she is in the early stages of Dementia, which for me was the most frustrating time. On hindsight there is no mileage in trying to convince your mum - I mean I even tried to convince her she had Dementia
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After a couple of years, there came a time when mum could not work the cooker or microwave or fill and switch the kettle on and that is the time we then called in the care (three times a day). I feel we may have done this a little later than we should have.
Depending on her condition, I feel the best way would be to introduce a carer gently into her household once a day and you to be there during those initial visits. Hopefully she will quietly accept the help when they make her a cup of tea and ask her to put her feet up and get into the routine as my mum did very quickly.
Maybe the best possible way to rationalise it to your mum is just tell her it is for your own peace of mind, you want to know she is safe on a daily basis and therefore ensure she can stay in her own home as long as possible. I have found the great fear and behind the obstinacy, was the thought of being sent to an old peoples home. Even now 9 years down the line, mum still tries to articulate how well she is doing in her own home, because, I guess, of that innate fear of being moved.
I wish you all the best in the journey you are commencing with your mum.