inductive hypothesis

[Noor01]

Hello,

I would like to ask a question that is related to the inductive hypothesis with Asymptotic notation.
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Example:
lg(n) = O(n)
Intuitively: lg(n) < n for all n ≥ 1.
Need a proof. Well
lg(n) < n ⇐⇒ n < 2n , for all n > 0.
Use induction on n to prove rhs.
 Base case n = 1 is clearly true.
 For induction step assume claim holds for n. Then
2n+1= 2 · 2n> 2n, by induction hypothesis.

To complete the proof just need to show that 2n ≥ n + 1.

Now
2n ≥ n + 1 ⇐⇒ n ≥ 1,
and we have finished.
So we take c = 1 and n0 = 1
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I don't know how we got this answer in these two lines:
1)2n+1= 2 · 2n> 2n
2)2n ≥ n + 1 ⇐⇒ n ≥ 1,

If you can explain this example for me, I would really appreciate that.

thxx