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## Homework Statement

Let x and y be real numbers. Prove that if x =< y + k for every positive real number k, then x =< y

## The Attempt at a Solution

x =< y + k

-y + x =< k

since k is positive, the lowest value it can take doesn't include 0: -y + x < 0

x < y

So I get x < y from x =< y + k and not the required x =< y. Am I right or I'm screwing up somewhere? Thanks for your help.