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The given problem asks to demonstrate, using the definition, that the sequence \( (u_n) \), defined by \( u_n = 20 \) for all \( n \in \mathbb{N} \), converges and to determine its limit.

### Solution:

1. **Sequence Definition**: The sequence \( u_n = 20 \) is a constant sequence, meaning for all values of \( n \), the value of the sequence is always 20. 

2. **Convergence**:
   - A sequence \( (u_n) \) converges to a limit \( L \) if for every \( \varepsilon > 0 \), there exists a natural number \( N \) such that for all \( n \geq N \), \( |u_n - L| < \varepsilon \).
   - In this case, since \( u_n = 20 \) for all \( n \), we can choose \( L = 20 \) as the limit.
   - For any \( \varepsilon > 0 \), we have:
     \[
     |u_n - L| = |20 - 20| = 0 < \varepsilon.
     \]
   - This inequality is true for any \( n \), so we can choose \( N = 1 \). Therefore, by the definition of convergence, the sequence \( (u_n) \) converges to 20.

3. **Limit**: The limit of the sequence is \( \lim_{n \to \infty} u_n = 20 \).

Thus, the sequence converges, and its limit is 20.

Would you like further details or have any questions?

Here are some related questions for practice:
1. What is the definition of a converging sequence in more general terms?
2. What are the properties of a constant sequence like \( u_n = 20 \)?
3. How does the choice of \( \varepsilon \) affect the convergence of a sequence?
4. Can a sequence have more than one limit? Explain why or why not.
5. How would the analysis change if the sequence was not constant?

**Tip**: In mathematics, constant sequences are the simplest examples of converging sequences, as they trivially converge to the constant value.

Soit (u_n) la suite définie pour tout n ∈ N par u_n = 20. En utilisant la définition, démontrer que la suite (u_n) converge et déterminer sa limite.

Learn how to prove that the constant sequence u_n = 20 converges, using the definition of sequence convergence. This problem is ideal for students learning about sequences, convergence, and limits, and is suitable for Grades 11-12.

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