How does the composition of functions prove this limit theorem?

[wizzerdo]

Prove the following theorem about the limit of the composition of functions.
Theorem 1 Let f : A → R and g : B → A. Suppose a is an accumulation point of
A and b is an accumulation point of B and that

i. lim t→b g(t) = a;
ii. there is a neighborhood Q of b such that for t ∈ Q ∩ B, g(t) NOT equal to a;
iii. limx →a f (x) = L.
Then f ◦ g has limit L at b.

 

[mathwonk]

i think if you just write down the definitions of the limit, look at your assumptions, and restate them in terms of the definition, this proof will write itself.