Limit of Sequence Whose Elements Depend on Previous Elements I was solving some exercises on limits, when I came across this problem:


  The point $C_1$ divides a segment $AB = l$ in half; the point $C_2$ divides a segment $AC_1$ in half; the point $C_3$ divides a segment $C_2C_1$ in half; the point $C_4$ divides $C_2C_3$ in half, and so on. Determine the limiting position of the point $C_n$ when $n \to \infty$.


I did successfully answer this particular exercise. However, it led me to a more general question.

How do you evaluate the limit of a sequence whose elements depend on prior elements? Sources on this topic are also welcome.

As another example, the Fibonacci sequence is one whose elements rely on prior elements (however, unlike the exercise above, there is no limit).

Note: I'd appreciate if nothing more advanced than limits are used. This exercise was found at the beginning of a calculus and analysis book, and all that I've covered so far are limits.
Show by induction:

$C_{2n}=C_2 ( \frac{3}{4})^{n-1}$ and $C_{2n+1}=C_1 ( \frac{3}{4})^{n}$ .

Hence $C_n \to 0$.
Show by induction:

$C_{2n}=C_2 ( \frac{3}{4})^{n-1}$ and $C_{2n+1}=C_1 ( \frac{3}{4})^{n}$ .

Hence $C_n \to 0$.