Is it possible to solve the following two problems on intervals using elementary methods, without using the outer measure ?

Problem 1

If $(I_n)$ is a disjoint sequence of subintervals of interval $I$ with $\bigcup_{n=1}^{\infty} I_n = I$, then $\sum_{n=1}^{\infty} l(I_n) = l(I)$, where $l$ denotes the interval length function.

I can see how to solve it where the $(I_n)$ are orderable :

Theorem

If $(I_n)$ is a disjoint sequence of subintervals of interval $I$ with $\bigcup_{n=1}^{\infty} I_n = I$, and if the $I_n$ can be rearranged into L to R order or into R to L order, then $\sum_{n=1}^{\infty} l(I_n) = l(I)$.

Proof

Consider the L to R case. Since the series has non-negative terms its convergence and sum are unaffected by rearrangements of its terms. So WLOG assume $(I_n)$ are in L to R order.

Case $I$ unbounded. Then $l(I) = \infty$. If $I_1$ unbounded then the result follows. Otherwise if $I_1$ is bounded then $I$ must have form $[a, \infty)$ or $(a, \infty)$, and LH of $I_1$ must be at $a$, and there must be no gaps between successive $I_n$, so that $l(I_1 \cup \cdots \cup I_n) = l(I_1) + \cdots + l(I_n)\ \forall\ n$. Given any $L > a, \exists\ N \mbox{ with } I_N \cap (L, \infty) \neq \emptyset$. Then $l(I_1 \cup \cdots \cup I_N) > L - a$, ie. $l(I_1) + \cdots + l(I_N) > L - a$. Thus partial sums are unbounded, and the result follows.

Case $I$ bounded. Consider the non-trivial case $l(I) > 0$. Then the $I_n$ are bounded also, and LH of $I_1$ must align with LH $a$ of $I$, and there must be no gaps between successive $I_n$, so that $l(I_1 \cup \cdots \cup I_n) = l(I_1) + \cdots + l(I_n)\ \forall\ n$. Because the $(I_n)$ are disjoint every partial sum is $\leq l(I)$. For any $\epsilon > 0, \exists\ N \mbox{ with } I_N \cap (a + l(I) - \epsilon, \infty) \neq \emptyset$. Then $l(I_1 \cup \cdots \cup I_N) > l(I) - \epsilon$, ie. $l(I_1) + \cdots + l(I_N) > l(I) - \epsilon$. Thus $\forall n \geq N, l(I) - \epsilon < l(I_1) + \cdots + l(I_n) \leq l(I)$. Thus the partial sums converge to $l(I)$.

The case of R to L follows similarly.

$\blacksquare$

However we cannot always rearrange a sequence of intervals $(I_n)$ into L to R or R to L order, eg. take $I = (0, 2)$ and form a sequence from the subintervals $\left(\frac{1}{n + 1}, \frac{1}{n}\right]$ for $n \geq 1$, the interval $(1, 1\frac{1}{2})$, and $\left[2 - \frac{1}{n}, 2 - \frac{1}{n + 1}\right)$ for $n \geq 2$ - this sequence then has no leftmost nor rightmost element.

Is there an elementary means of proving the general case, where no ordering can be assumed, or is outer measure necessary to solve this problem ?

Problem 2

Prove that if $\delta > 0$ and if $(E_n)$ is a sequence of sets in $[0, 1]$ with each $E_n$ containing a finite number of non-overlapping closed intervals with a total length $\geq \delta$, then there exists a point belonging to infinitely many of the $E_n$.

This result is readily proved using Axler [1, §2.60, p44] on the measure of a decreasing intersection, applied to outer measure on $\mathbb{R}$ on the Borel sets, since we require to prove that the set $E = \bigcap_{n=1}^{\infty} \bigcup_{m=n}^{\infty} E_m$ is non-empty, and we have from that theorem $|E| = \lim_{n \rightarrow \infty} |\bigcup_{m=n}^{\infty} E_m|$, and $\forall\ n,\ |\bigcup_{m=n}^{\infty} E_m| \geq |E_n| \geq \delta$, so that $|E| > 0$, as required.

Problem 2 is part of a proof of the Bounded Convergence Theorem for Riemann Integrals in Bartle [2, §22.14, p288], but Bartle omits proof, though the rest of the proof of the BCT is provided. In Bartle & Sherbert [3, §7.2.5, p304], the BCT for Riemann Integrals is stated but the proof is omitted entirely and described as 'quite delicate'. Bartle's book [2] covers quite a bit of real analysis but does not anywhere mention the concept of outer measure. It would be interesting to know if an elementary solution exists for Problem 2 or not. I asked about Problem 1 on MSE but did not receive any answers.

[1] Sheldon Axler (2020), Measure, Integration & Real Analysis, Springer Graduate Texts in Mathematics, https://measure.axler.net/.

[2] Robert G. Bartle (1964), The Elements of Real Analysis, John Wiley & Sons.

[3] Robert G. Bartle & Donald R. Sherbert (1982), Introduction to Real Analysis, John Wiley & Sons.