Exercises 4

1. Prove the Triangle inequality:
   If a, b are real numbers then | a + b | ≤ | a | + | b |.
   When is this inequality strict ?
   Why is it called the Triangle inequality ?
   Use induction to prove that if a₁, a₂, ..., a_nare any real numbers then |a_i| ≤ |a_i|.

   Solution to question 1

2. Prove the Squeeze rule:
   If a_n≤ x_n≤ b_n for all n and the sequences (a_n) and (b_n) converge to the same limit, then (x_n) also converges to this limit.

   Solution to question 2

3. Give examples or prove the non-existence of sequences which are:
   (a) convergent but not monotonic,
   (b) bounded but not convergent,
   (c) convergent but not bounded,
   (d) monotonic but not bounded,
   (e) divergent to +∞ but not monotonic,
   (f) monotonic and bounded but not convergent,
   (g) unbounded but not monotonic,
   (h) positive and convergent, but not monotonic.

   Solution to question 3

4. The sequence ( (2n + 1)/(n + 1) ) converges to 2. How big must we choose N to be so that the terms of the sequence are within ε of this limit if ε = 0.1, 0.01, 0.0001 ?

   Do the same thing for the sequence ( (2n²+ 1)/(n²+1) ).

   Solution to question 4

5. Give examples or prove the non-existence of sequences such that:
   (a) (|a_n|) converges but (a_n) does not converge,
   (b) (a_n) converges but (|a_n|) does not converge,
   (c) (a_n) and (b_n) do not converge, but (a_n+ b_n) converges,
   (d) (a_n) and (a_n+ b_n) converge but (b_n) does not converge.

   Solution to question 5

6. Let a be a real number in (0, 1). Let a_n be the real number obtained by deleting the first n digits of the decimal expansion of a.
   For example, if a = 0.1234567... then a₁ = 0.234567.., a₂ = 0.34567..., etc.
   For which a is the sequence (a_n) convergent and what is its limit ?

   Solution to question 6