Homework 3 for MATH 104
Due: Tuesday, September 26, 9:30am in class
Problem 1
Given a sequence (sn)n∈N, let (−sn)be the sequence defined as (−s1,−s2,−s3, . . . ).
Show that lim infn→∞(sn)=−lim supn→∞(−sn).
Problem 2
Let (qn)n∈Nbe a sequence. Suppose that there exists an r∈R, 0 <r<1 such that for
all n∈N,
|qn+2−qn+1|6r|qn+1−qn|.
Show that (qn)is a Cauchy sequence.
(Remark: You will have to use a fact about the convergence of the sequence rn.
You are required to prove this fact.)
Problem 3
A real number xis called algebraic if it is the solution of a polynomial with integer
coefficients, i.e. if there exists a natural number nand integers an,an−1, . . . , a1,a0with
an6=0 such that
anxn+· · · +a1x+a0=0.
Show that the set of all algebraic real numbers is countable.
Problem 4
Show that the set R×R={(x,y) : x,y∈R}has the same cardinality as R.
(Remark: This implies that the plane has the same cardinality as the real line!)
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