Worksheets — Set 17
Introductory topology · Separation and countability
Print and write your answers. Use diagrams to show thinking.
Warm-up
1. Prove: the sum of two even integers is even. ___ 2. Give a direct proof that √20 is irrational (outline key step). ___ 3. Prove by induction: 1 + 2 + … + n = n(n+1)/2
_See printable PDF for diagram._
Written practice
1. Prove: the sum of two even integers is even. ___ 2. Give a direct proof that √19 is irrational (outline key step). ___ 3. Prove by induction: 1 + 2 + … + n = n(n+1)/2 4. State the contrapositive of: if n² is even, then n is even ___ 5. Find a counterexample if false: every continuous f is differentiable 6. Write a ε-δ style argument outline for a limit. ___ 7. Give a direct proof that √22 is irrational (outline key step). ___ 8. Give a direct proof that √25 is irrational (outline key step). ___
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More written practice
1. Give a direct proof that √28 is irrational (outline key step). ___ 2. Give a direct proof that √31 is irrational (outline key step). ___ 3. 4 + 7 = ___ 4. 5 + 4 = ___ 5. 6 + 5 = ___ 6. 2 + 6 = ___
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Patterns and rules
1. Prove: the sum of two even integers is even. ___ 2. Give a direct proof that √36 is irrational (outline key step). ___ 3. Prove by induction: 1 + 2 + … + n = n(n+1)/2
_See printable PDF for diagram._
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Stretch
1. Prove: the sum of two even integers is even. ___ 2. Give a direct proof that √44 is irrational (outline key step). ___
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Parent tip: Start from what is given