This paper investigates convergence behavior of composition sequences f 1 ∘ f 2 ∘ … ∘ f n ( z ) and f n ∘ f n − 1 ∘ … ∘ f 1 ( z ) where the f n 's are bilinear transformations and f n → z . Additional results are provided for the case when the f n 's are more general functions.