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docnet

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- Homework Statement
- Consider the following statements about real numbers x and y. If true, prove it. If false, state its negation and prove that statement.

- Relevant Equations
- 1. ##\forall x>0, \exists y>0## such that ##xy<1##.

2. ##\exists y>0## such that ##\forall x>0, xy<1##.

1. True Let ##y=\frac{1}{x+1}##. Then ##xy=\frac{x}{x+1}<1, \forall x>0##.

2. False. Let ##x=\frac{1}{y}##. Then for any ##x>0##, you always have ##xy=\frac{y}{y}=1##

I feel like the second statement is wrong because I am choosing any ##y## , instead of choosing any ##x##.

2. False. Let ##x=\frac{1}{y}##. Then for any ##x>0##, you always have ##xy=\frac{y}{y}=1##

I feel like the second statement is wrong because I am choosing any ##y## , instead of choosing any ##x##.

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