In a poset $P = (S,\leq)$, we say that a chain is any sequence $x_1,x_2,\ldots,x_k$ of distinct elements in $S$,such that $x_1\leq x_2\leq\ldots\leq x_k$.
For example, consider the poset $P = (P(A),\subseteq)$, where $A = \{1,2,\ldots,n\}$. One example of a chain here could be the sequence $\emptyset$, $\{2, 5\}$, $\{1, 2, 5\}$, $\{1, 2, 3, 4, 5\}$, because $$\emptyset\subseteq \{2, 5\} \subseteq \{1,2,5\} \subseteq \{1,2,3,4,5\}.$$ So, the question is... What is the largest number of elements you can have in a chain, in the poset $P = (P(A),\subseteq)$ defined in this problem? Prove that your claim is correct.