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The problem given is written in Persian, and I will translate and solve it:

### Translation:
"Assume \( f \) and \( g \) are increasing functions, and at least one of them is strictly increasing. Prove that:
\[
f(x) \leq g(x) \quad \forall x \in \mathbb{R}
\]
implies
\[
f(f(x)) \leq g(g(x)) \quad \forall x \in \mathbb{R}.
\]"

### Solution Outline:
1. \( f \) and \( g \) are increasing functions, which means if \( x_1 \leq x_2 \), then:
   \[
   f(x_1) \leq f(x_2) \quad \text{and} \quad g(x_1) \leq g(x_2).
   \]
   
2. We are given \( f(x) \leq g(x) \) for all \( x \in \mathbb{R} \).

3. Since \( f(x) \leq g(x) \), applying \( f \) and \( g \) to both sides preserves the inequality, because both functions are increasing. Thus:
   \[
   f(f(x)) \leq f(g(x)).
   \]
   
4. Similarly, since \( f(x) \leq g(x) \), and applying \( g \) to both sides (again due to the increasing nature of \( g \)):
   \[
   g(f(x)) \leq g(g(x)).
   \]
   
5. We need to show that \( f(f(x)) \leq g(g(x)) \), which follows directly from the above steps since \( f(f(x)) \leq f(g(x)) \leq g(g(x)) \).

Thus, the desired inequality is proved.

---

Would you like more details on any step of this proof?

Here are some related questions for further practice:
1. What changes if both \( f \) and \( g \) are strictly increasing?
2. Can you provide a counterexample if neither \( f \) nor \( g \) are increasing?
3. How does the proof change if \( f \) and \( g \) are only non-decreasing instead of strictly increasing?
4. Can you extend this result to compositions of more than two functions?
5. What if \( f \) and \( g \) were decreasing functions instead?

**Tip:** When dealing with increasing functions, inequalities are preserved when applying the function to both sides.

Assume f and g are increasing functions, and at least one of them is strictly increasing. Prove that: f(x) ≤ g(x) for all x ∈ ℝ implies f(f(x)) ≤ g(g(x)) for all x ∈ ℝ.

Learn how to prove that f(f(x)) ≤ g(g(x)) when f and g are increasing functions, with at least one strictly increasing, and f(x) ≤ g(x). This advanced calculus problem involves key concepts like monotonicity and function composition.

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