Problem Set 3

  1. Easy
    Let \(\left\{x_{k}\right\}\) and \(\left\{y_{k}\right\}\) be sequences such that \(x_{k}\to a\) and \(y_{k} \to b\). Show that \(x_{k}+y_{k}\to a+ b\) and \(x_{k}y_{k}\to ab\).

  2. Easy
    Let \(x_{k} = \frac{3k+4}{k-5}\). Given \(\epsilon > 0\), find an integer \(K\) such that \(\vert x_{k}-3\vert < \epsilon\) for all \(k > K\).

  3. Easy
    Let \(\left\{x_{k}\right\}_{k=1}^{\infty}\subseteq\mathbb{R}^{n}\) be a sequence, and let the components be given by \( x_k = (x_k^1, x_k^2, \ldots, x_k^n) \). Prove that \(x_k \to x = (x^1,x^2,\ldots,x^n)\) if and only if \(x_{k}^{i}\to x^{i}\) for all \(i\). In other words, a sequence in \(\mathbb{R}^{n}\) converges if and only if each of its components converges.

  4. Medium
    Find an example of a sequence \(\left\{x_{k}\right\}\) such that \(\vert x_{k+1} - x_{k} \vert \to 0\), but \(\left\{x_{k}\right\}\) is not Cauchy.

  5. Medium
    Let \(S\subseteq\mathbb{R}\), and set \(L = \inf S\). Show there exists a sequence \(\left\{x_k\right\}\) converging to \(L\).

  6. Medium
    Prove whether each of the following sequences converges or does not have a limit
    1. \(((-1)^k, 0)\)
    2. \((\cos(\pi k/2), \sin( \pi k/2))\)
    3. \(\left(\frac{\cos\pi k}{k},\frac{\sin \pi k}{k}\right)\)
    4. \(\left( (1+1/k)\cos(\pi k/2), (1+1/k)\sin(\pi k/2)\right)\)
    5. \((\cos(Ck),\sin(Ck))\) where \(C/\pi\) is irrational [Hard]
    6. \((k^2, 0, 1/k)\)

  7. Medium
    If \(\left\{x_{k}\right\}\subseteq\mathbb{R}^{n}\) is a Cauchy sequence and there exists a subsequence \(\left\{x_{k_{n}}\right\}\) such that \(x_{k_{n}}\rightarrow x\), then \(x_{k} \to x\).

  8. Hard
    Construct a sequence \(\left\{z_{k}\right\}_{k=1}^{\infty}\subseteq \mathbb{R}^{2}\) with the property that for any point \(z = (x,y)\) such that \(x^2+y^2 = 1\), there exists a subsequence \(\left\{z_{k_{n}}\right\}_{n=1}^{\infty}\) such that \(z_{k_{n}}\to z\).

  9. Hard
    Construct a sequence \(\left\{x_{k}\right\} \subseteq \mathbb{R}^{2}\) with the property that for any \(x \in \mathbb{R}^{2}\), there exists a subsequence \(\left\{x_{k_{n}}\right\}\) which converges to \(x\).

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