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I want to prove that there exists a positive number $x$ such that $x - \dfrac{2}{x} <1$ and $x \leq 2$.

Starting a proof by contradiction, I assumed that there is a positive number x such that x - (2/x) > 1 and x <= 2. I multiplied the first inequality by x and got x^2 - x - 2 > 0. I'm confused where to go from here and how to use the fact that x must also be less than or equal to 2.

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    $\begingroup$ The easiest way to prove existence is to explicitly produce one example. In this case it's not too hard to consider $x = 1$ - it obviously satisfies both conditions. There is nothing left to prove. $\endgroup$ – mathguy Sep 6 '18 at 4:06
  • $\begingroup$ For contradiction, you should assume that for all positive $x$ with $x \leq 2$, we have $x - \frac 2x \geq 1$. Then, you should show that this cannot happen. Simplify this to $x^2 -x- 2 \geq 0$, which goes to $(x+1)(x-2) \geq 0$. Now, the product of two numbers is positive if and only if both have the same sign, so $0 < x \leq 2$ and this equation cannot simultaneously have a solution. $\endgroup$ – астон вілла олоф мэллбэрг Sep 6 '18 at 4:08
  • $\begingroup$ @астонвіллаолофмэллбэрг But it is necessary to distinguish the cases $x\ge 0$ and $x<0$ when we multiply an inequality with $x$. $\endgroup$ – Peter Sep 6 '18 at 7:07
  • $\begingroup$ @Peter $0 < x \leq 2$ is part of the assumptions, hence the simplification goes through. $\endgroup$ – астон вілла олоф мэллбэрг Sep 6 '18 at 7:55
  • $\begingroup$ @астонвіллаолофмэллбэрг You are right, I overlooked the word "positive" $\endgroup$ – Peter Sep 6 '18 at 7:56
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Take ANY positive number such that:

$$x<2\tag{1}$$

This implies:

$$\frac2x>\frac{2}{2}$$

$$\frac2x>1$$

$$-\frac2x<-1\tag{2}$$

Add (1) and (2) and you get:

$$x-\frac2x<1$$

Actually all positive numbers less than 2 are solutions to your problem. The only exception is $x=2$.

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You made a error in "I assumed that there is a positive number x such that x - (2/x) > 1 and x <= 2."

To use proof by contradiction, it should be "For all positive x, x - (2/x) > 1 or x <= 2."

Actually the original statement is wrong. There does NOT exist a positive number x such that x - (2/x) > 1 and x <= 2.

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