Put the steps to the proof that the sum t = r + s is irrational if r is a rational and s is a rational in the correct sequence

To prove that the sum \( t = r + s \) is irrational under the assumption that both \( r \) and \( s \) are rational, it seems there may be confusion. In fact, the correct statement is:

If \( r \) and \( s \) are both rational numbers, then their sum \( t = r + s \) is also rational.

However, if you meant to prove that the sum \( t = r + s \) is irrational under the assumption that at least one of them is irrational, here are the series of steps for that proof:

1. **Define Rational and Irrational Numbers**: Recall that a rational number can be expressed as \( \frac{a}{b} \) where \( a \) and \( b \) are integers and \( b \neq 0 \). An irrational number cannot be expressed in this form.

2. **Assume Rational and Irrational**: Let \( r \) be a rational number, which can be written as \( \frac{a}{b} \), and let \( s \) be an irrational number.

3. **Express the Sum**: Consider the sum \( t = r + s = \frac{a}{b} + s \).

4. **Rearranging**: Rearranging gives us \( s = t - \frac{a}{b} \).

5. **Assume \( t \) is Rational**: If \( t \) were rational, then both \( t \) and \( \frac{a}{b} \) would be rational.

6. **Conclude that \( s \) is Rational**: If \( t \) is rational and \( \frac{a}{b} \) is rational, then \( s = t - \frac{a}{b} \) would also have to be rational because the set of rational numbers is closed under subtraction.

7. **Contradiction**: This contradicts the assumption that \( s \) is irrational.

8. **Final Conclusion**: Therefore, the original assumption that \( t = r + s \) is rational must be false, and we conclude that \( t = r + s \) is irrational if \( r \) is rational and \( s \) is irrational.

Here's the sequence summarized:
1. Define rational and irrational numbers.
2. Assume \( r \) is rational, \( s \) is irrational.
3. Consider \( t = r + s = \frac{a}{b} + s \).
4. Rearrange to get \( s = t - \frac{a}{b} \).
5. Assume \( t \) is rational.
6. Conclude \( s \) must be rational.
7. Find a contradiction.
8. Conclude \( t = r + s \) is irrational.

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