- #1

Mr Davis 97

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## Homework Statement

Let ##a,b \in \mathbb{R}##. Show if ##a \le b_1## for every ##b_1 > b##, then ##a \le b##.

## Homework Equations

## The Attempt at a Solution

We will proceed by contradiction. Suppose that ##a \le b_1## for every ##b_1 > b##, and ##a > b##. Let ##b_1 = \frac{a+b}{2}##. We see that ##b_1>b##: ##a>b \implies \frac{a+b}{2}> b \implies b_1 >b##. Hence, by the hypothesis, it must be true that ##a \le b_1##. But since ##a \le b_1 \implies a \le \frac{a+b}{2} \implies a \le b##, we have reached a contradiction. We simultaneously have that ##a \le b## and ##a >b##. Hence, it must be the case that the original statement is true, and we are done.