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please solve these questions with explanation

Here are the solutions to the questions in the provided image:

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### **31.**
**Statement**: Poljubna vsota dveh padajočih zaporedij je padajoče zaporedje.  
**Solution**:  
This statement is **false**. Adding two decreasing sequences does not necessarily result in a decreasing sequence. For instance, if the sequences decrease at different rates, their sum may not decrease consistently. Counterexample: \( a_n = 1/n \), \( b_n = -1/n^2 \). Both are decreasing, but their sum \( c_n = a_n + b_n \) is not decreasing.

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### **32.**
**Statement**: Vsako strogo naraščajoče zaporedje ima limito.  
**Solution**:  
This statement is **false**. A strictly increasing sequence does not always have a finite limit. For example, the sequence \( a_n = n \) is strictly increasing but diverges to infinity, so it does not have a finite limit.

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### **33.**
**Statement**: Padajoče zaporedje realnih števil je divergentno natanko takrat, ko je navzgor omejeno in navzdol neomejeno.  
**Solution**:  
This statement is **true**. A decreasing sequence of real numbers diverges only when it is bounded above and unbounded below. In this case, it tends to \(-\infty\).

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### **34.**
**Natančno formuliraj izrek o sendviču.**  
**Solution**:  
The **Sandwich Theorem** (Izrek o sendviču) states:  
If \( a_n \leq b_n \leq c_n \) for all \( n \geq N \) (for some integer \( N \)), and  
\[
\lim_{n \to \infty} a_n = \lim_{n \to \infty} c_n = L,
\]
then  
\[
\lim_{n \to \infty} b_n = L.
\]

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### **35.**
**Statement**: Če je funkcija \( f: \mathbb{R} \to \mathbb{R} \) padajoča, potem obstaja največ eno število, ki je lahko limita rekurzivnega zaporedja \( x_1, x_2, x_3, \dots \), podanega s pravilom \( x_{n+1} = f(x_n) \).  
**Solution**:  
This statement is **true**. If \( f \) is a decreasing function, then it is injective (one-to-one), meaning that the sequence can converge to at most one limit. If a fixed point \( L \) exists such that \( f(L) = L \), this is the unique limit.

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### **36.**
**Statement**: Če je zaporedje \( x_1, x_2, x_3, \dots \) konvergentno in je funkcija \( f: \mathbb{R} \to \mathbb{R} \) odvedljiva, potem je tudi zaporedje \( f(x_1), f(x_2), f(x_3), \dots \) konvergentno.  
**Solution**:  
This statement is **true**. If \( \{x_n\} \) converges to \( L \), and \( f \) is differentiable, then \( f(x_n) \to f(L) \) by continuity of differentiable functions. Hence, \( f(x_n) \) is also convergent.

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Let me know if you would like any of these solutions explained in more detail or if you have additional questions!

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### **5 Related Questions**:
1. What happens when two strictly increasing sequences are added? Is the result always increasing?
2. Can a bounded, decreasing sequence ever diverge? Why or why not?
3. How is the Sandwich Theorem applied to prove the convergence of sequences?
4. What are the implications of a function being injective in the context of recursive sequences?
5. How does differentiability of a function influence the convergence of related sequences?

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### **Tip**:
When analyzing sequences, always check their boundedness and monotonicity; these properties are critical in determining convergence.

Solve questions related to limits, sequences, and series in the provided image, including the Sandwich Theorem and properties of sequences.

Explore detailed solutions to problems involving limits, sequences, and series, including the Sandwich Theorem, monotonicity, and convergence. Perfect for undergraduate students learning advanced calculus.

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