Can the Relativistic Velocity Equation Keep Velocities Below the Speed of Light?

[Ayame17]

Homework Statement

From the relativistic law of composition of velocities one deduces that if a particle has velocity ui in O, then in a system O' moving at -vi relative to O, its velocity will be u'i where:

u'=(u+v)/(1+(uv/c^2))

Show, mathematically, that if 0<u<c and 0<v<c, then 0<u'<c also.

Homework Equations

Hint: Think of above equation as (u'/c)=[((u/c)+(v/c))/(1+(uv/c^2))]

The Attempt at a Solution

Well, I've looked at it, and figured that if 0<u<c, then 0<(u/c)<1 and the same with v and u'. However, when putting this into the form of the 'hint' equation, you'd get u'/c=(0<2)/(1<2) which would give 0<u'<2, which isn't right. I can't see where it's going wrong though. Any help?

 

[Ayame17]

Sorry about the mess with the brackets, I couldn't get Latex to work how I wanted it!