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To prove the theorem stated in the image, the formal expression states that:

Let \( l \in \mathbb{R} \). Then \( l = \lim X_n \) if and only if:

1. For every \( \epsilon > 0 \), there exists an \( n \in \mathbb{N} \) such that \( X_k > l - \epsilon \) for all \( k \geq n \).

### Proof:
This is essentially a variant of the definition of the limit of a sequence in real analysis.

We need to show that \( l = \lim X_n \) if and only if the condition described holds. Here's how we can break it down:

### Part 1: \( l = \lim X_n \Rightarrow \) the given condition holds.

#### Hypothesis:
- By the definition of a limit, \( l = \lim_{n \to \infty} X_n \) means that for every \( \epsilon > 0 \), there exists an \( n \in \mathbb{N} \) such that:
  \[
  |X_k - l| < \epsilon \quad \text{for all} \quad k \geq n.
  \]
  This implies:
  \[
  l - \epsilon < X_k < l + \epsilon \quad \text{for all} \quad k \geq n.
  \]
  From this, we directly get:
  \[
  X_k > l - \epsilon \quad \text{for all} \quad k \geq n.
  \]
Thus, the condition is satisfied.

### Part 2: The given condition holds \( \Rightarrow l = \lim X_n \).

#### Hypothesis:
- We are given that for every \( \epsilon > 0 \), there exists an \( n \in \mathbb{N} \) such that \( X_k > l - \epsilon \) for all \( k \geq n \).

We now want to show that this implies \( l = \lim X_n \). 

We already know that \( X_k > l - \epsilon \) for all \( k \geq n \), but to fully prove this, we would need to extend this to also show the upper bound \( X_k < l + \epsilon \) (i.e., the sequence stays within an \( \epsilon \)-neighborhood of \( l \)).

Thus, combining both conditions (lower and upper bounds), we conclude that \( X_k \) converges to \( l \), and hence \( l = \lim X_n \).

### Conclusion:
We've shown that both directions hold, proving that \( l = \lim X_n \) if and only if the given condition about \( X_k \) and \( l - \epsilon \) holds.

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Would you like more detailed steps or clarification on any part of this proof?

Here are some related questions for further exploration:

1. What is the general definition of a sequence converging to a limit in \( \mathbb{R} \)?
2. Can we extend this theorem to sequences in other metric spaces?
3. How would the proof change if we considered \( X_n \) to be a sequence of complex numbers?
4. What happens if the sequence \( X_n \) does not converge to \( l \)? Can we still apply a similar condition?
5. How does this theorem relate to the Cauchy criterion for convergence?

**Tip**: When proving limits, always carefully track both lower and upper bounds to ensure the sequence is fully contained within the \( \epsilon \)-neighborhood of the limit.

Prove the theorem: Let l ∈ R, Then l = lim X_n if and only if: 1.) given ε > 0, there exists n ∈ N such that X_k > l - ε for all k ≥ n.

Learn how to prove a theorem related to the limit of a sequence in real analysis. This problem involves the epsilon-delta definition of limits and requires understanding key concepts in mathematical limits.

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