2 Hints
As per usual, here’s where you’ll find the problem sheet hints!
Think about all the tests for convergence that you’ve seen so far! A reminder for d) — the growth factor test won’t work on a ratio of polynomials.
See the hint for question 1 (and the one on the sheet!) Another reminder — if you use a test or a theorem anywhere, state it!
Firstly, this is an if and only if statement. So you have to prove two things:
- \((a_n)\) convergent \(\implies\) \((a_{2k})_k, (a_{2k-1})_k\) and \((a_{3k})_k\) are convergent.
- \((a_{2k})_k, (a_{2k-1})_k\) and \((a_{3k})_k\) are convergent \(\implies\) \((a_n)\) convergent.
For the second implication, try and find common subsequences of the three given subsequences to show that \((a_{2k})_k, (a_{2k-1})_k\) and \((a_{3k})_k\) all have the same limit. One of the tutorial questions from Problem Sheet 7 might come in handy at this point.
- Can you think of any inequalities relating \(k\) and \(n_k\), where \(k \in \mathbb{N}\)? Also, this question asks about monotone sequences, so you’ll have to prove that this result holds for both increasing and decreasing sequences.
- Again, two things to prove here (due to the if and only if). Make sure you understand the definitions involved — it’ll make it easier to construct the proofs.
Start by unpacking the definitions in the question, and try to make one look like the other.
This follows a similar procedure to a proof involving the definition of convergence. Fix \(\epsilon > 0\) and show that for \(n\geq m\) we can bound \(\lvert a_n - a_m \rvert\) above by \(1/2^{m-1}\). Then, note (or prove by induction) that \(m \leq 2^m\) to find a upper bound which is easier to deal with, and use this to find the \(N \in \mathbb{N}\) in the Cauchy sequence definition. You’ll also find useful the triangle inequality and geometric series formulae here, but make sure to state when you use them!