To show that the sequence \( a_n = 2 - \frac{1}{n} \) converges, we need to find its limit as \( n \) approaches infinity.

### Step 1: Identify the Sequence
The sequence is given by:
\[
a_n = 2 - \frac{1}{n}
\]

### Step 2: Compute the Limit
As \( n \) approaches infinity, the term \( \frac{1}{n} \) approaches 0. Therefore, the sequence can be analyzed as follows:
\[
\lim_{n \to \infty} a_n = \lim_{n \to \infty} \left(2 - \frac{1}{n}\right)
\]
\[
\lim_{n \to \infty} a_n = 2 - \lim_{n \to \infty} \frac{1}{n}
\]
Since \( \lim_{n \to \infty} \frac{1}{n} = 0 \), we have:
\[
\lim_{n \to \infty} a_n = 2 - 0 = 2
\]

### Step 3: Conclusion
The sequence \( a_n = 2 - \frac{1}{n} \) converges, and its limit is 2.

Would you like more details, or do you have any other questions?

Here are some related questions:
1. What is the definition of the limit of a sequence?
2. How does one prove that a sequence is bounded?
3. What are some other examples of sequences that converge to a constant?
4. How can you determine if a sequence diverges?
5. What is the difference between convergence and divergence?

**Tip:** When working with sequences, always check if the terms approach a specific value as \( n \) becomes large to determine convergence.

Learn how to show that the sequence 2 - 1/n converges using mathematical proof. Understand the concept of sequences and limits, and follow a step-by-step explanation of the solution. Suitable for advanced high school students and anyone interested in mathematical analysis.

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