By induction according to the size of the base set \( X \).
If it is single-element, then this element is the largest.
Choose any \( a \in X \). From the inductive hypothesis, \( (X \setminus a, \leq) \) has the largest element \( b \). If \( a > b \), then \( a \) is the largest element on \( (X, \leq) \) from the transitivity, otherwise \( b \) remains the largest element even on \( (X, \leq) \), because \( a \) cannot be incomparable with \( b \) due to the linearity of the order.