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Less than or equal relation is transitive.

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

2 comments

  1. Jay says:
    • Jay says:

      Hey Rori, sorry i am not very familiar with latex code
      i typed up an alternate soln in Lyx and cut and pasted its contents here.
      Any ideas how to integrate the symbols?

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