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

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.

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.

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 $-\infty$.

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
$\lim_{n \to \infty} b_n = L.$

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.

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.

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

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?

Tip:

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