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 $n > 2$.

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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.