Let an be a bounded sequence and bn such that
the limit bn as n→∞ is b and
0<bn ≤ 1/2 (bn-1)
Prove that if:
an+1 ≥ an - bn,
The Attempt at a Solution
as 0<bn ≤ 1/2 (bn-1) the sequence bn is decreasing.
thus its limit, b, is 0 (can i assume that?)
Then, assuming an converges to a limit, say p, the equation is:
p ≥ p - b where b=0
p ≥ p is true for every p.
Then, an is constant and therefore converges.
Does is it work like that?