Proof: F (x) is monotonically increasing, if there is {xn} - > + ∞, such that limf (xn) = a (n - > ∞), then limf (x) = a (x - > ∞) It is proved that let limf (xn) = a For any E > 0, there exists N1 such that | f (xN1) - a | + ∞, there exists a natural number n. when n > N, there is xn > xN1 Then | f (xn) - f (xN1) | ≤ | f (xN1) - A| The last step is to use Heine's theorem

Proof: F (x) is monotonically increasing, if there is {xn} - > + ∞, such that limf (xn) = a (n - > ∞), then limf (x) = a (x - > ∞) It is proved that let limf (xn) = a For any E > 0, there exists N1 such that | f (xN1) - a | + ∞, there exists a natural number n. when n > N, there is xn > xN1 Then | f (xn) - f (xN1) | ≤ | f (xN1) - A| The last step is to use Heine's theorem

For any E > 0, there exists xN1 such that | f (xN1) - A|

X0 = a, X1 = B, xn = 1 / 2 (XN-1 + xn-2) prove the convergence of XN and find its limit value

It can be seen from the meaning of the title: X (n) > 0, (n > = 0)
We have: X (n) = 1 / 2 (x (n-1) + X (n-2)) x (n-1) = 1 / 2 (x (n-2) + X (n-3)) and so on, and add them all to get:
X (n) + 1 / 2x (n-1) = 1 / 2x (0) + X (1) = 1 / 2A + B, n > 2, then x (n) is bounded
The limit is 1 / 3A + 2 / 3B

It is proved that the function f: I → R is continuous at XO ∈ I, any xn ∈ I, xn → XO (n →∞), and has constant LIM (n →∞) F（
Xn)=f(Xo

If it is continuous at x0, it is obvious that limf (xn) = f (x0). Conversely, since xn tends to x0, then when n is sufficiently large | xn-x0|