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## Homework Statement

Prove the following propositions:

1) ∀x ∈ (0, 1), ∃y ∈ (0, 1), x < y

and

2) ∀x, y ∈ R, if x < y, then ∀b ∈ (0, ∞), ∃a ∈ (0, ∞),

x + ab < y.

Can anyone help me out with either one?

I have a few others that I can get but I can't get these two. Mainly because these don't have a specific method to use so I don't know which to use. Where as previously the ones said prove by contradiction etc.

Methods that can be used:

Direct Proof

Contraposition

Contradiction

## Homework Equations

1) ∀x ∈ (0, 1), ∃y ∈ (0, 1), x < y

and

2) ∀x, y ∈ R, if x < y, then ∀b ∈ (0, ∞), ∃a ∈ (0, ∞),

x + ab < y.

## The Attempt at a Solution

How would I know with method to use for these?