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You're replying to a comment by Jason Yeo.
There are two steps to the method.
1. Guess the solution
2. Prove the solution by induction
For example, we try to get the upper bound of T(n) = T(n/2) + 1 (this is the recurrence relation for binary search)
My guess is o(lgn) so our guess should look like T(n) <= clgn for some c and n > N.
Then we prove the guess by induction.
We assume that T(k) <= clgk holds for k < n.
T(n) = T(n/2) + 1
<= clg(n/2) + 1 (note: we invoke the assumption here)
= clgn - clg2 + 1
= clgn - (clg2 - 1) (note: we rearrange them to the form desired - residual)
<= clgn when c >= 1/(lg2)
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