Complicated proofs aside, there's a simple way to prove this on a coordinate plane:
Draw a vertical line going through (0,0). Think about the definition of a slope, $\Delta$Y/$\Delta$X. The change in Y ($\Delta$Y) can be anything, depending upon how far up the line you go - so let's call $\Delta$Y n.
Now, the change in X ($\Delta$X) is zero. No matter how high up on the line you get - no matter what the value of Y is - $\Delta$X is always 0.
So, we can say that the slope of a vertical line is n/0. But is that undefined? In a way, it is, so your brother isn't technically wrong. But it can also be thought of as infinity:
Take a diagonal line with slope 1/1. 1/1 is the same as 1. Now, take a diagonal line with a steeper slope - let's say 2/1. 2/1 is the same as 2. If we take a steeper line, we find that $\Delta$Y/$\Delta$X gets bigger and bigger, until it approaches infinity - or, in other words, the limit of $\Delta$Y/$\Delta$X approaches infinity. In this way, the slope of a vertical line can be thought of as infinity - because once you get to that vertical line, the slope is infinity.
Side Note: This is how my math teacher showed that anything divided by 0 equals infinity - the slope of a straight line is both n/0, undefined, AND infinity, thus showing that n/0 equals infinity.
Presumably, your brother is thinking that the slope is infinity. You're actually both right - it's both undefined and infinity. You should decide for yourself how you want to respond to the question and your brother, but I think your safest bet is to explain that the slope is both undefined and infinity. However, you might want to say that the slope is infinity and leave it there - saying that it is undefined has the connotation of saying that it has no slope. That's obviously not true, but your instructor could take it that way.
Side Note #2: It's probably too late for this to help anything - almost two years late - but I hope this clears up any controversy or doubt you had in your mind as to what the slope of a vertical line is.
Hope this helped!