- #1

pixelate

- 3

- 0

## Homework Statement

Hi, I've been solving Calculus Deconstructed by Nitecki and I've been confused by a particular lemma in the book. Namely:

If a sequence is eventually bounded, then it is bounded:

that is, to show that a sequence is bounded, we need only find a number

γ ∈ R such that the inequality

[tex]|x_i| < \gamma \ \text{holds for all i ≥ K, for some K}[/tex]

## Homework Equations

3. The Attempt at a Solution

[/B]

However if we consider the sequence ##\tan {\frac{\pi}{n}} ##, with n starting at 1. For terms when n is very high, we can find some value for ##\gamma## for which the condition holds and thus the sequence is bounded (according to the lemma). Yet at n=1, ##\tan {\frac{\pi}{n}} = \infty##, so the sequence cannot be bounded.

My questions are:

1. Is my reasoning at all correct in the first place ?

2. It is easy to see that my sequence does have a lower bound but no upper bound, so does the lemma only refer to one bound (But doesn't bounded mean bounded in both directions?)

Thanks for reading!