If she holds my arm like I'm a proper gentleman, I have one arm/hand free to open doors, etc. But if she were really about to fall..... I wouldn't easily catch her w/o a big shift, b/c my hands are in front and to side of her. If I put one arm around her back for full support, then my other arm is.. well, she can grab my hand that side, but is not as stable as my gentleman's arm would be, plus if I have both hands full of her, I can't be opening doors etc. ????
She doesn't really need that much support.... ie she doesn't need a walker instead of a cane. But when we go to doctors etc and she is unsure, that translates into hesitance w walking, and automatic 'don't think I can do this', blah blah (every. Single. Time. We. Go out and she doesn't want to.) OH. Mom has mild dimentia and is almost 95.
If mom is resistant, and you are both insistent on walking together, with you as her support, holding on to your one arm is giving her the support she needs to feel more comfortable, as long as you are strong enough to support her weight with your one arm.
If she needs more support than that, it is dangerous for both of you if you think you are going to catch her or stop her from falling. She really needs a chair for safety if she is unsteady.
These things might look to you like something that she can immediately use with your guidance, but without proper training for you and for her, disaster is possible. Ask her doctor about prescribing physical and/or occupational therapy and make sure you attend with her. Since she has dementia (mild or not), don't depend on her remembering much. You drive the bus now.
I hope it goes well.
I'll bet she'd like it.
A lot of seniors use Walker/Rollators for balance, and not always for support. They make life much easier. And, you can still walk beside her as a gentleman and assist her. Except with a Walker, she can focus more on chatting with you - and less on worrying about falling or losing her balance.
Go to google: type in "How to assist elder with walking". You can access films also.
Much depends here on the anxiety (usual regarding possibility of falls) the elder has. Sometimes this makes a person unable to properly follow instructions, and it may take some practice (kinda like dancing!).