MATH 310 — SPRING 2007
EXAM I SOLUTIONS
1. (9 points) Negate each of the following statements. (Feel free to use the symbols ∀,∃, “s.t.”,
etc.)
(a) For all real x,x6= 0 ⇒x2>0.
Negation: There exists an x∈Rsuch that x6= 0 but x2≤0.
(b) For every real M, there exists a real number Nsuch that, for every n>N,|f(n)|> M.
Negation: There exists a real number Msuch that, for every real N, there exists an
n>N such that |f(x)| ≤ M.
(c) For all ∈P, there is a δ∈Psuch that, for all x, y ∈R,
|x−y|< δ ⇒ |f(x)−f(y)|< .
Negation: There exists an ∈Psuch that, for every δ∈P, there exist x, y ∈Rsuch
that |x−y|< δ but |f(x)−f(y)| ≥ .
2. (10 points) We proved in class the following:
FACT:
n
X
i=1
i=n(n+ 1)
2for all n∈N.
Use this fact and induction to prove that
n
X
i=1
i3= n
X
i=1
i!2
for all n∈N.
Proof. We prove this statement by induction on n.
Basis Step: If n= 1,
n
X
i=1
i3= 1 and n
X
i=1
i!2
= 1. X
Induction Step: Suppose that
k
X
i=1
i3= k
X
i=1
i!2
for some k∈N. We want to show that
k+1
X
i=1
i3= k+1
X
i=1
i!2
.
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