Yes, remove the old limb, it is not helping the over health of the plant. The plants maybe spending energy trying to keep this frond alive, when it has no chance of recovery.
It appears to me that this is a glassed in deck. I believe that once the frond reach a certain height they go from partially shaded to full blast of the sun. Glass intensives light ray making them much stronger. The brown parts look like a water issue. It might be that frond was losing water through transpiration quicker than the plant could pump water to it. Young palms love protection for the hot sun. This is a very young palm. In the wild it would still be growing under larger palm, never getting the full blast of the suns rays. You are giving it intensive ray.
Move the palm to a different location in the room. That might protect the new leaves from ending up with the same situation.
My last bit of knowledge, which might have something to do with why your older leaves are so yellow. Palms require different amounts of nutrients than other trees and shrubs, like higher amounts of Magnesium. You can add magnesium by adding magnesium sulphate, which you can buy at the garden centre or buy some epsom salts. Magnesium helps the plant to take up other nutrients like iron. People often try nitrogen and iron to green up their palms, which will help, but only if the additional magnesium is in place. Apply fertilizer only when the plant is actively growing. Iron should be used sparingly. Read the directions and follow what they box says when applying iron and other nutrient. The easiest thing is buy Palm Fertiliser. They have the extra nutrients your palm needs.
I hope I was able to help.