Suppose we have a set $M = (0,1) \subset R$ of reals well-ordered as the first uncountable ordinal.

Let $M(a) = \lbrace x \in M : x < a \rbrace$. For every $a \in M$ set $M(a)$ is countable. That's why every increasing sequence is bounded:
$$(*) ~~~~~~~~~~ \forall \lbrace a_1,...,a_n,...\rbrace \subset M ~~\exists b \in M : a_i < b ~~\forall i \in \mathbb{N}.$$

Now suppose that we can pick elements from $M$ at random. And let's try to build an increasing random sequence by the following algorithm. Let we have an increasing sequence of elements $\{a_1,...,a_n\}$. Pick some random number $b$. If $b > a_n$ set $a_{n+1} = b$. Otherwise pick other random number instead of $b$ and check $b > a_n$ condition. Continue this till success. Since we pick numbers at random it shouldn't be a problem to construct an infinite sequence. But every infinite sequence is bounded! Which means that in our infinite process we will never be able to pick random number which is greater than some number $c \in M$. This is even more astonishing since $M(c)$ is countable and $M \setminus M(c)$ is uncountable!

Any thoughts how to "solve" this paradox?

Suppose we have a set $M = (0,1) \subset R$ of reals well-ordered as the first uncountable ordinal.

Let $M(a) = \lbrace x \in M : x < a \rbrace$. For every $a \in M$ set $M(a)$ is countable. That's why every increasing sequence is bounded:
$$(*) ~~~~~~~~~~ \forall \lbrace a_1,...,a_n,...\rbrace \subset M ~~\exists b \in M : a_i < b ~~\forall i \in \mathbb{N}.$$

Now suppose that we can pick elements from $M$ at random. And let's try to build an increasing random sequence by the following algorithm. Let's say we have an increasing sequence of elements $\{a_1,...,a_n\}$. Pick some random number $b$. If $b > a_n$ set $a_{n+1} = b$. Otherwise pick other random number instead of $b$ and check $b > a_n$ condition. Continue this till success. Since we pick numbers at random it shouldn't be a problem to construct an infinite sequence. But every infinite sequence is bounded! Which means that in our infinite process we will never be able to pick random number which is greater than some number $c \in M$. This is even more astonishing since $M(c)$ is countable and $M \setminus M(c)$ is uncountable!

Any thoughts how to "solve" this paradox?