Example 3 " $x>1$ " is " $x \geqslant 1$ " ( ).
(A) Sufficient but not necessary condition
(B) Necessary but not sufficient condition
(C) Sufficient and necessary condition
(D) Neither sufficient nor necessary condition
Solution: Let $A=\{x \mid x>1\}, B=\{x \mid x \geqslant 1\}$. Clearly, $A \subseteq B$, but $B \nsubseteq A$, that is, if $x \in A$, then $x \in B$, and $1 \in B$, but $1 \notin A$. Therefore, “$x>1$” is a sufficient but not necessary condition for “$x \geqslant 1$”. Choose A.

Key Points for Solving: Using the inclusion relationship between sets to determine what kind of condition one statement is for another is essentially another way of expressing the implication relationship. Generally, if the set of sufficient conditions for a conclusion is $A$, the set of necessary and sufficient conditions is $B$, and the set of necessary conditions is $C$, then $A \varsubsetneqq B \varsubsetneqq C$, as shown in the figure.