# Bounded Monotonic Sequence Theorem

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1. Jan 26, 2016

### Euler2718

1. The problem statement, all variables and given/known data

Use the Bounded Monotonic Sequence Theorem to prove that the sequence:

$$\{a_{i} \} = \Big\{ i - \sqrt{i^{2}+1} \Big\}$$

Is convergent.

2. Relevant equations

3. The attempt at a solution

I've shown that it has an upper bound and is monotonic increasing, however it is to my understanding that for me to use this theorem the sequence must be bounded (and of course have monotonicity) - i.e. have an upper and lower bound. I can't seem to show that this has a lower bound; even graphically, it just blows off to negative infinity. How should I proceed?

2. Jan 26, 2016

### Staff: Mentor

Why do you think this? You didn't say anything about the possible values of i, but I assume they are positive integers {1, 2, 3, ...}.

Consider $f(x) = x - \sqrt{x^2 + 1}$. All of the values of this function are negative, since $x < \sqrt{x^2 + 1}$ for all x > 0. As x gets larger, the difference is smaller, and approaches zero. The largest difference comes when x is smallest (i.e., closest to zero).

3. Jan 26, 2016

### Ray Vickson

If $f(i) = i -\sqrt{i^2+1}$ is monotonically increasing, then $f(1) < f(2) < f(3) < \cdot$, so if $f(1)$ is a finite number, it is a lower bound!

4. Jan 26, 2016

### Euler2718

Thank you, I think I got it now.