Monotonic convergence

Question

Using monotonic convergence, prove that the sequence is convergent (no need to find the limit):

a) (1*3*5*...*(2n-1))/(2*4*6*...*(2n))

b)  (1 − (1/10)) * (1 − (1/(10)^2) ) * ... * (1 − (1/10^n) ).

I understand the theorem, but how do I neatly prove for these sequences

Answer 1

a) We need to start the sequence at n=1, and the first term=1/2. The next term is ⅜ so the sequence is decreasing. Since no term can be less than zero, we can conclude that the sequence has a lower bound of zero. The sequence seems to be bounded below, so it will converge to that value, according to the monotone convergence theorem.

b) If we plug in n=0, we get (theoretically) zero. With n=1 we get 0.9; with n=2 we get 0.891, and the next value in the sequence will be even smaller because each term introduces a factor which is less than 1. The sequence is bounded below (no term in the sequence cam be negative), and the theorem states that the infimum (zero) will be the convergence value.