NAME
CR 8
MAT 300 · SPRING 2015
Critique the following proof, as if you were providing detailed feedback on someone’s
homework. Write your comments directly on this sheet.
Also give a revised version of the proof. Your revision should use the basic ideas of the
original proof (if possible), but it should be an improvement in terms of reasoning and
exposition. Write your proof on the back of this sheet.
Problem. Suppose that P (n) is a statement for each integer n ≥ 1, and:
(a) P (1) is true,
(b) P (n) → P (2n) for each integer n ≥ 1, and
(c) P (n) → P (n − 1) for each integer n ≥ 2.
Prove that P (n) is true for all n ∈ N.
Proof.
For each n ∈ N, let Q(n) = P (2n ). In other words, Q(n) is true if
and only if P (2n ) is true. Then by (a), we know that Q(0) = P (20 ) = P (1)
is true. Now let n ∈ N be arbitrary, and suppose Q(n) is true. Then by
definition, P (2n ) is true, and it follows that Q(n + 1) = P (2n+1 ) = P (2 · 2n ) is
also true. This shows that Q(n) is true for all n ∈ N by induction. In other
words, P (2n ) is true for all n ∈ N.
Next, let n ≥ 2 be arbitrary, and suppose P (n) is true. Then P (n − 1) is
true by (c), and since we know P (2) is true from the case n = 1 above, it
follows by induction that P (n) is true for all n ≤ 2.
All together, this shows that P (n) is true for all n ∈ Z. and a standard
induction argument.
Date: April 13, 2015. Due Date: Thursday, April 30, 2015.
S. Kaliszewski, School of Mathematical and Statistical Sciences, Arizona State University.