So, I don't know where I suddenly became some kind of MATH WIZZUM. It's an interesting question that I'm sure has been explored, but your proof makes mathematical if not logical sense. Another way of proving this would be, I believe, to make a graph y=xth root 2 or whatever. I don't know if that produces a continuous graph, but if there was a vertical asymptote over 0, that would also prove this.
Also, in the vein of general bullstuffting about school, I took Math III in eighth grade, didn't do my homework and got a B or some stuff confining me to Algebra I for 9th grade. I then took Geometry over the summer online and am now in Algebra II.
Online courses are awesome. I did a semester of Geometry in two weeks.