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4. Midterm 1 Review

≪ 3. Sequences and Limits | Table of Contents | 5. Midterm 1 Review Solutions ≫

Due to popular demand, I’ve decided to structure this week’s discussion around reviewing for the midterm. I have a small collection of test-taking advice if you struggle with testing in proof-based math classes!

Here is a collection of problems that I think would be good for practise. I may or may not come back and upload solutions to them. They’re primarily focused on proofs, and you can find lots of practise with computation in the textbook.

You should know all of the definitions we have covered so far by heart! If someone shakes you awake in the dead of night and demands the formal definition of a limit, you should be able to answer them.

Problem 1.

Let \(S\) and \(T\) be bounded sets of real numbers such that their intersection \(S\cap T\) is nonempty. Prove that \[\begin{align*} \sup \left( S\cap T \right) \leq \min \left\lbrace \sup S, \sup T \right\rbrace. \end{align*}\] Give an example where they are not equal.

Problem 2.

Let \(a, b\) be nonzero real numbers, and let \(c, d \) be arbitrary real numbers. Prove that \[\lim _{n\to\infty} \frac{an+c}{bn+d} = \frac{a}{b}.\]

Problem 3.

Let \(S\) and \(T\) be bounded sets of nonnegative real numbers. Define the set \[S\times T := \left\lbrace s t : s\in S, t\in T\right\rbrace,\] and show that \[\sup \left( S\times T \right) = \sup(S) \sup(T). \]

Problem 4.

Give an example of two disjoint sets \(S\) and \(T\) such that \(\sup S = \sup T\).

Problem 5.

Give an example of a bounded sequence of real numbers \(\left( x_n \right) _{n=1}^{\infty}\) such that the sequence \[S_n = \frac{1}{n}\left( x_1+x_2+\cdots+x_n \right)\] does not have a limit. Prove your answer.

Problem 6.

Let \(\left( x_n \right) _{n=1}^{\infty}\) be any bounded sequence of real numbers. Show that \(M = \limsup _{n\to\infty} x_n\) if and only if for every \(\epsilon > 0\), there exist infinitely many values of \(n\) for which \(x_n > M - \epsilon \).

Problem 7.

Define the sequence \(\left( x_n \right) _{n=1}^{\infty}\) recursively as follows: \[\begin{align*} x_1 &= 0, && \\ x _{n} &= \sqrt{x _{n-1} + 1} && \forall n > 1. \end{align*}\]

  1. Show that the sequence is monotone, i.e. \(x_n \leq x _{n+1}\) for all \(n\).
  2. Show that the sequence is bounded, i.e. there exists some real number \(M\) such that \(x_n \leq M\) for all \(n\).
  3. Conclude that the sequence has a limit.
Hint
Try using induction for \(1\) and \(2\).

Challenge Problem 8.

We say that a sequence of vectors \(\mathbf{v}_n \in \mathbb{R}^2\) converges to a limit \(\mathbf{L}\in \mathbb{R}^2\) if for all \(\epsilon > 0\), there exsits some \(N_ \epsilon \) such that for all \(n > N_ \epsilon \), \(\left\lVert \mathbf{v}_n - \mathbf{L} \right\rVert < \epsilon \). Here, \(\left\lVert (x, y) \right\rVert = \sqrt{x^2 + y^2}\) is the usual Euclidean norm.

Prove that a pair of sequences \(\left( x_n \right)\) and \(\left( y_n \right)\) converge to limits \(X\) and \(Y\) respectively if and only if the sequence of vectors \(\left( x_n, y_n \right)\) converges to \(\left( X, Y \right)\).

Does this generalise to \(\mathbb{R}^d\)?