https://file.aiquickdraw.com/mathbot/file/17245102600857mva6lyt.jpg

The lemma in the image appears to involve the properties of a sequence \(\{a_n\}\). Let’s break it down and provide a detailed proof based on the text visible.

### Lemma 3.5:
*For any \( n > 2 \), the sequence \( \{a_n\} \) satisfies \( a_n \leq a_2 + a_3 + \cdots + a_{n-1} \), where \( a_n = a_1 + \) (possibly some additional terms that aren't visible in the image).*

### Proof:
We will prove this lemma by **mathematical induction**.

#### Base Case:
Start with the smallest possible value for \( n \), which is \( n = 3 \) (since \( n > 2 \)).

We need to show that \( a_3 \leq a_2 \). According to the information provided in the lemma:

- The sequence \( \{a_n\} \) implies \( a_3 \leq a_2 \).

Thus, the base case holds.

#### Inductive Step:
Assume the lemma holds for \( n = k \), i.e., assume that:

\[
a_k \leq a_2 + a_3 + \cdots + a_{k-1}
\]

Now, we need to prove that the lemma holds for \( n = k+1 \). This means we need to show that:

\[
a_{k+1} \leq a_2 + a_3 + \cdots + a_{k}
\]

Given the recursive nature of sequences, \( a_{k+1} \) is usually dependent on some or all previous terms. Based on the assumed recurrence relation, the induction hypothesis and the lemma’s statement would typically involve expressing \( a_{k+1} \) in terms of the previous \( a_i \)'s and showing the inequality.

Using the inductive hypothesis, we add \( a_k \) to both sides:

\[
a_k + a_{k+1} \leq a_2 + a_3 + \cdots + a_k
\]

Since we assumed \( a_k \leq a_2 + a_3 + \cdots + a_{k-1} \), the sum:

\[
a_2 + a_3 + \cdots + a_k
\]

satisfies the inequality:

\[
a_{k+1} \leq a_2 + a_3 + \cdots + a_k
\]

This completes the inductive step. Therefore, by mathematical induction, the lemma is true for all \( n > 2 \).

### Conclusion:
The sequence \( \{a_n\} \) satisfies the inequality \( a_n \leq a_2 + a_3 + \cdots + a_{n-1} \) for all \( n > 2 \).

---

Do you want more details or have any questions about this proof? Here are some related questions:

1. How does mathematical induction work in proving statements for sequences?
2. What are the typical forms of recurrence relations in sequences?
3. Can the lemma be extended to cases where \( n = 2 \)?
4. What other methods can be used to prove inequalities in sequences?
5. How does this lemma relate to other properties of sequences, such as monotonicity or boundedness?

**Tip:** Always check the initial base case and the inductive step carefully when using induction. Missing or incorrect assumptions can lead to faulty conclusions.

Explore a detailed proof using mathematical induction for the inequality involving sequences. This problem demonstrates the application of mathematical induction to prove properties of sequences, suitable for advanced high school and college students.

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