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Let $c_0(\mathbb{N})$ be the space of real-valued sequences converging to zero.

Here, each element $\{\alpha_n\} \in c_0(\mathbb{N})$ itself is a "set", so that we can think about set-theoretic inclusion among different elements of $c_0(\mathbb{N})$.

That is, an element $\{\beta_n\}$ of $c_0(\mathbb{N})$ can be a "subset" of another element $\{ \alpha_n\}$, WITHOUT necessarily being a subsequence

Now, let us define a maximal element of $c_0(\mathbb{N})$ to be one that is not properly contained in any other element in the set-theoretic sense described above.

Then, I wonder what would be the cardinality of maximal elements in $c_0(\mathbb{N})$. Existence of a maximal element is guaranteed by Zorn's lemma, I believe. However, I am curious about "how many" they are.

My guess is that they are at most countable, but I cannot see how to prove or disprove...Could anyone please help me?

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    $\begingroup$ There are no maximal (in this sense) elements of $c_0$. Given a sequence, $\alpha_n\to 0$, you can always shift it one unit to the right and then include an extra value that wasn't in $\alpha_n$ to start with. $\endgroup$
    – Christian Remling
    Commented Jun 10, 2023 at 16:58
  • $\begingroup$ Oh....I see. Thank you for your insight.. $\endgroup$
    – Isaac
    Commented Jun 10, 2023 at 17:00

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