- #1

- 2,571

- 1

## Homework Statement

I already have the solutions, just don't understand one step

Let [tex](n_k)_{k=1}^{\infty}[/tex] be a sequence of integers

[tex]n_1 < n_2 < n_3 < ...[/tex]

Then [tex]n_k \geq k[/tex] [tex] \forall k = 1,2, ...[/tex]

In particular [tex]n_k \to \infty[/tex] when [tex]k \to \infty[/tex]

Prove this is true

**2. The solution**

Employ Induction

(1) Base Case: k = 1

[tex]n_1 \geq 1[/tex]

(2) Induction: Assume [tex]n_k \geq k[/tex] for the case k. We show k + 1 case

[tex]n_{k+1} > n_k > k \implies n_{k+1} > k \implies n_{k+1} \geq k + 1[/tex]

Hence the result holds

**Question**

How on earth did we get to [tex]n_{k+1} \geq k + 1[/tex] from [tex] n_{k+1} > k [/tex]