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**1. Homework Statement**

Prove that for each [tex]n \in N[/tex] (aka natural numbers), [tex]2^n \geq n+1[/tex]

Prove that for each [tex]n \in N[/tex] (aka natural numbers), [tex]2^n \geq n+1[/tex]

## Homework Equations

**3. The Attempt at a Solution**

Let the proposition P(n) be "[tex]2^n \geq n+1[/tex]"

Clearly P(n) is true for n=1, [tex]2^1 \geq 1+1[/tex].

We suppose P(k) is true, i.e., supposing that [tex]2^k \geq k+1[/tex] is true, then,

[tex]2^{k+1} \geq (k+1)+1[/tex]

I think it can then be rewritten as [tex]2.2^{k} \geq 2.(k+1)[/tex]. Does anyone know the next step? I'm not sure what to do from here...

Thanks!

Let the proposition P(n) be "[tex]2^n \geq n+1[/tex]"

Clearly P(n) is true for n=1, [tex]2^1 \geq 1+1[/tex].

We suppose P(k) is true, i.e., supposing that [tex]2^k \geq k+1[/tex] is true, then,

[tex]2^{k+1} \geq (k+1)+1[/tex]

I think it can then be rewritten as [tex]2.2^{k} \geq 2.(k+1)[/tex]. Does anyone know the next step? I'm not sure what to do from here...

Thanks!