If $A$ is subset of $B$ and $B$ is a subset of $C$ , then prove that $A$ is subset of $C$ .

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Solution

Let $A$ is a subset of $B$ and $B$ is a subset of $C$ .
Then there is an $x$ in $B$ such that, for every $x$ in $B, x$ is in $C$ and there exists an $x$ in $A$ such that, for every $x$ in $A, x$ is in $B$ . Thus, for every $x$ in $A, x$ is also in $C$ , and thus $A$ is a subset of $C$ .
Hence proved.

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