You don't know if or are positive. But you can take in the equation, go back to the equation and take the derivative on both side, then take .
Let be the functions defined by and , where . Prove that and are linearly independent in .
For our assumption to be true, the values of and are dependent on the value of , therefore there is no universal value of or , except , for which
Thus, these functions are linearly independent
This proof is correct?
is the 0 function, not just the number 0. this means this equation holds for any and all values of t.
in particular, it holds for t = 0, whence b = -a.
thus we have for all t. if a is not 0, then
for all real t, taking logs of both sides gives r = s, a contradiction, so a = 0 and b = -a = 0.
EDIT: why do i do it this way? because in the original poster's proof, he takes the log of a number which he is trying to prove is 0.
now, by the previous two comments, you can actually assume that a > 0, and b < 0, and then the original poster's argument works,
but it is not easy to see that ln(a/-b) = t(s - r) immediately implies a contradiction. i feel it is more straight-forward to use the fact that R
is a field (and thus has no zero divisors).