Zorn's Lemma states that if every chain $C$ in a partially ordered set $X$ has an upper bound then there is at least one maximal element in $X$.
Let $n\in \mathbb{N}$, $C_n \subset \mathbb{N}$ be the set containing the natural numbers $\{1,\dots,n\}$. Then if $k\le n$, $C_k\subset C_n$, so $C_n$ is a chain. Also, $C_n$ has an upper bound, such as $n+1$. But what is the maximal element of $\mathbb{N}$?
There's something I don't get in Zorn's Lemma statement.
You just take some examples of chains, but there are many others, for example the multiples of any positive integer is a chain. The hypothesis of the lemma is that every chain must have a upper bound, but $\mathbb{N}$ is a total ordered set, then every subset of $\mathbb{N}$ is a chain, in special, $\mathbb{N}$ itself is a chain.
The Zorn's lemma is quite useless in a total ordered set, because, if this set satisfies the hypothesis, the upperbound of the set itself (which is a chain) is the maximal element.