Lemma. For any infinite sequence of numbers, there exists an infinite non-decreasing subsequence. Proof. This is evident as if the sequence is bounded, one number appears infinitely many times, forming an infinite non-decreasing subsequence, while if it is unbounded one can choose progressively larger numbers. We show the following: Stronger version of problem wrote: For any infinite set $S$ of numbers in base $b\ge 2$, there exists an infinite subset of numbers $\{a_1,a_2,\cdots\}$,for which $a_i$ is contained in $a_{i+1}$ for all $i\ge 1$. Call such a subset "good". Proof. We induct on $b$. Base case $b=1$ is trivial by the lemma. Consider some base $n$. We construct a subset as follows: pick anything as $a_1$ and, if possible, pick $a_{i+1}$ such that $a_i$ is contained it. If we can keep going, we have found the desired subset. Else we get stuck at some number $a_x=T$. Let the digits of $T$ be $d_1,d_2,\cdots, d_k$. For each of the remaining numbers, we highlight the following digits, scanning from left to right: The first occurrence of $d_1$, the first occurrence of $d_2$ after that, etc. By assumption we never reach $a_k$. By pigeonhole there is some $m$ such that for infinitely many of these numbers we only highlight $m$ digits. These numbers must be in the form .....$d_1$....$d_2$..... etc .....$d_m$...... The first dotted section has no $d_1$, the second has no $d_2$, etc, the last has no $d_{m+1}$. We can thus use the induction hypothesis to find an infinite sequence satisfying the condition in the first dotted section, and in that pick an infinite subset satisfying the condition in the second dotted section, etc, and we are done. Thank you to user Phorphyrion for pointing out my mistake.

This is also a special case of Kruskal's tree theorem. The set of labels is the digits $0, 1, \dots, 9$ with the discrete partial order (any digit can only be compared with itself). Each positive integer corresponds to a labeled line graph, and the root is taken to be the leftmost vertex. Any infinite set corresponds in this way to a set of rooted labeled trees so that by the Theorem two trees are inf-embeddable into each other. But this exactly corresponds to one number being contained in the other. BTW the proof above might generalize to a proof of Kruskal's theorem.