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## Homework Statement

Let x and y be real numbers such that x<y. There exists z that is a real number such that x<z<y.

## Homework Equations

## The Attempt at a Solution

I wrote the following: We are given that x<y. We know from previous proof that (1/2)<1.

Consider y such that y=x+1

x+1<x and we know (1/2)<1. Therefore x+(1/2)<x+1

Also, x<x+(1/2). Therefore x<x+(1/2)<x+1 which shows the existence of a z such that x<z<y.

For this problem my professor marked me wrong at the part where I said let y=x+1. He just draws a red line and leaves it up to us to figure what we did wrong. I do not see how this argument is invalid though. Can someone explain?