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The image contains a mathematical problem written in French. Here is the transcription:

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**Exercice 5** Pour tout entier \( n \geq 1 \), on pose \( E_n = \{k + \frac{k}{n} : k \in \mathbb{N}^* \} \).

a) Montrer que l'ensemble \( E_n \) a une borne inférieure et que \( \inf E_n = \inf \{k + \frac{k}{n} : 1 \leq k \leq n\} \).

b) Montrer que l'on a toujours \( \inf E_n \geq 2 \sqrt{n} \). Quand y a-t-il égalité?

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This exercise involves the analysis of a set \( E_n \), defined in terms of natural numbers and an expression involving \( k + \frac{k}{n} \). It asks you to show that this set has a lower bound and to explore the conditions under which the infimum is greater than or equal to \( 2 \sqrt{n} \), as well as identifying when equality holds.

Would you like me to walk through the solution process for part (a) or (b) of this exercise?

### Relevant follow-up questions:
1. What is the infimum of a set and how is it calculated?
2. How do we show that a set has a lower bound?
3. What mathematical techniques are used to find the infimum of \( E_n \)?
4. How can inequalities involving \( \inf E_n \) and \( 2 \sqrt{n} \) be proved?
5. What role do natural numbers play in determining the properties of the set \( E_n \)?

### Tip:
For problems involving infimum and sets, focus on the structure of the set and how its elements behave as \( n \) changes.

Exercice 5: Pour tout entier n >= 1, on pose E_n = {k + k/n : k ∈ N*}. a) Montrer que l'ensemble E_n a une borne inférieure et que inf E_n = inf {k + k/n : 1 ≤ k ≤ n}. b) Montrer que l'on a toujours inf E_n >= 2√n. Quand y a-t-il égalité?

Explore how to find the infimum of the set E_n = {k + k/n : k ∈ N*} and prove the inequality inf E_n ≥ 2√n. This problem dives into real analysis concepts, such as sets, bounds, and inequalities.

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