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A woman that is 78 years old has probably been through more that we can imagine. She's probably had to fight her whole life to prove that she is strong, and carrying a cane could be to her a sign of weakness. I'm sure by this point, she knows she isn't young, and will never be young again, but maybe carrying a cane may make her feel older than she is willing to feel right now. She may look on it as giving up or giving in to old age.
You're right, but people are stubborn. Maybe if she had a nice cane, something wooden, or different she would feel better about carrying it. Eventually she will start using it, it's just going to take time. Tell her how good she looks with it, and maybe that might help. :)