how that each of the following sequences {ai} is monotonic(or has a monotonic tail) and is bounded, and therefore converges by the Bounded Monotonic Sequence Theorem.

Question

anonymous · Answer

$$\frac{ i }{ i^2 +1 }$$

anonymous · Answer

Would it be decreasing?

sirm3d · Answer

prove that either $a_i\ge a_{i+1}$ (monotonically decreasing) or $a_i\le a_{i+1}$ (monotonically decreasing) for all positive integers $i$. my bet is on the first case (decreasing).
assume the inequality and show that the solution to the inequality is all positive integers $i$.

anonymous · Answer

wait so I did it out and found out it was decreasing but like what does that mean? Does that show that it is monotonic?

sirm3d · Answer

because it is decreasing, is is bounded above by the first term.

anonymous · Answer

Oh okay thanks

sirm3d · Answer

oh, don't forget
$$\frac{1}{2}\ge a_i>a_{i+1}>0$$ so the sequence converges to zero.