A fellow student of mine asked a question to our teacher in functional analysis, and the answer we got was not very satisfactory. In our discussion on Banach spaces the student asked "Why is it interesting/important for a normed space to be complete?". To my surprise the teacher said something along the lines; "well, we want sequences where the elements become very close to have a limit in the space". What my fellow student and I was hoping for, was some deeper reason or explanation for why we are interested in complete normed spaces. Some deep theorems or results that require that a space is complete, for instance. Does anyone have something to add to this discussion?