Tag: Transitivity

Less than or equal relation is transitive.

by RoRi

June 30, 2015

Prove that if $a \leq b$ and $b \leq c$ , then $a \leq c$ .

Proof.
By the transitivity of $<$ (Theorem I.17), we have that if $a < b$ and $b < c$ , then $a < c$ .
Then, if $a = b$ and $b \leq c$ we have $a \leq c$ by substitution.
If $b = c$ and $a \leq b$ , then $a \leq c$ by substitution.
If $a = b$ and $b = c$ , then $a = c$ by transitivity of the $=$ relation. Hence, $a \leq c$ by definition of $\leq$ .
Thus, in all cases $a \leq b$ and $b \leq c$ implies $a \leq c. \qquad \blacksquare$