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The problem asks to prove by induction that \( 2^n > n \) for all \( n \in \mathbb{N} \) (the set of natural numbers).

### Step-by-step proof using induction:

#### Base Case (n = 1):
We first check if the inequality holds for \( n = 1 \).
\[
2^1 = 2 \quad \text{and} \quad n = 1.
\]
Clearly, \( 2 > 1 \), so the base case is true.

#### Induction Hypothesis:
Assume that for some \( n = k \), the inequality holds, i.e.,
\[
2^k > k.
\]
This is the induction hypothesis.

#### Induction Step:
We need to prove that if the inequality holds for \( n = k \), then it also holds for \( n = k + 1 \). That is, we need to show that:
\[
2^{k+1} > k+1.
\]

Starting with the left-hand side:
\[
2^{k+1} = 2 \cdot 2^k.
\]
By the induction hypothesis, we know that \( 2^k > k \), so:
\[
2^{k+1} = 2 \cdot 2^k > 2 \cdot k.
\]
Thus, we now need to show that \( 2k \geq k + 1 \). Simplifying this inequality:
\[
2k \geq k + 1 \quad \Longleftrightarrow \quad k \geq 1.
\]
This is true for all \( k \geq 1 \).

Therefore, the induction step holds, and by the principle of mathematical induction, we have proven that \( 2^n > n \) for all \( n \in \mathbb{N} \).

---

Would you like further details or clarifications? Here are some related questions:

1. What is the principle of mathematical induction and why is it useful?
2. Can you apply this induction method to other exponential functions, like \( 3^n > n^2 \)?
3. What happens if we test the inequality \( 2^n \geq n \) for \( n = 0 \)?
4. How does the inequality \( 2^n > n \) behave as \( n \) becomes very large?
5. How do we modify the proof if \( n \in \mathbb{Z^+} \) (positive integers)?

**Tip:** Inductive proofs often work well when the statement to be proved involves recursive or iterative structures.

Prove by induction that 2^n > n for all n in N (natural numbers).

A detailed proof by mathematical induction to show that 2^n is greater than n for all natural numbers. This proof covers the base case, induction hypothesis, and induction step, providing a clear understanding of inequalities and exponents. Suitable for advanced high school students and undergraduates.

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