- #1

B3NR4Y

Gold Member

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- 8

## Homework Statement

1. Let ε > 0. Determine how large n ∈ ℕ must be to ensure that the given inequality is satisfied, and use the Archimedean Property to establish that such n exist.

a.) [itex] \frac{1}{n} < \epsilon [/itex]

b.) [itex] \frac{1}{n^{2}} < \epsilon [/itex]

c.) [itex] \frac{1}{\sqrt{n}} < \epsilon [/itex]

b.) [itex] \frac{1}{n^{2}} < \epsilon [/itex]

c.) [itex] \frac{1}{\sqrt{n}} < \epsilon [/itex]

## Homework Equations

The archmidean property says that ∀ε > 0, ∀M > 0, ∃n ∈ ℕ such that n*ε > M

## The Attempt at a Solution

For part a I multiplied both sides by n, which made the inequality 1 < n*ε, which is a statement of the archimedian property with M = 1, so in order for this to always be true, n = 2, but I run into problems with the reasoning of that, because ε > 0, it says nothing about it being an element of ℕ, so I'm not sure if that's right.

For C I noticed that squaring both sides, becomes [itex] \frac{1}{n} < \epsilon^{2} [/itex], but redefining ε

^{2}as δ, we have the same statement as part a. But I'm still stuck because I'm not sure if the question wants a numerical answer or what.

For 2 I know the answer is "We can use any value of δ satisfying the double inequality 0 < δ < [itex] \frac{\epsilon}{2} [/itex]", but I'm not even sure where to start getting this.