answer as true or false if true explain why if false provide a counter example ( do not correct a
false statement)
a) let (a_n)_neN be a sequence that possesses a convergent subsequence. Then (a_n)_neN is
bounded
b) let (an)nEN and (bn)nEN be sequences that both diverge. Then (a_n + b_n)_neN is also
divergent.
c) let x> 0 Then x + (1/x) > = 2
Solution
a. FALSE. Consider teh sequence an = n if n is even
= 1/n if n is odd.
Clearly the odd subsequence is convergent but the sequence itself is not bounded.
b. FALSE. Let an = n for all n,
bn = -n for all n. Clearly both diverge. But their sum is the constant sequence cn =
an+bn = 0 which is obviously converget.
c. TRUE.
x + 1/x - 2 = (x2- 2x + 1)/x = (x-1)2/x>0 since teh numerator is a square and teh denominator is
positive by assumption.