Instructions

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Teamwork is encouraged but every person must turn in their own assignment. Please reference who you worked with and all external sources used. External help (not full solutions) or information is permitted but must be acknowledged. Lecture notes, OH help, course books, recorded lectures are not considered external and can be used without reference.

When solving a point of a problem you may give for granted any problem or point that appears prior to it, including problems on previous HW sheets (even if you didn't manage to solve them) but must referenced (e.g. "I use the proven fact that…" or "I use the fact from HW1P2 that…").

All statements require proof, unless specified otherwise.

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1

1.1

Show that given any \(x\in\R\), \(x>0\) and any \(p\in\N\) there exists a unique \(y\in\R\), \(y>0\) such that \(y^{p}=x\), this is called \(\sqrt[p]{x}\).

Hint: it is \( \sup \{y\in\R\colon 0\leq y^{p}\lt x\} \)

1.2 ​   hardish

Let \(a_{n}\) be a non-negative sequence in \(\Q\) that satisfies

\begin{equation*} a_{n}\leq \frac{a_{n-1}+2a_{n-2}}{4}. \end{equation*}

Show that \(a_n\) has a limit and find it.

1.3 ​   very_hard optional

Let \(a_{n}\) be a non-negative sequence in \(\R\) that satisfies

\begin{equation*} a_{n}\leq \frac{a_{n-1}+a_{n-2}}{2}. \end{equation*}

Show that is has a limit.

2

Let us endow \(\R\) with a different metric:

\begin{equation*} d^*(x,y) = \Big|\frac{x}{1+|x|}-\frac{y}{1+|y|} \Big| \end{equation*}

2.1

Show that \((\R,d^*)\) is a metric space.

Hint:

  • Do not bring the fractions in the definition of \(d^*\) to a common denominator.
  • Do not forget to prove all properties of metrics (not all of them are immediate).

2.2

Show that \((\R,d^*)\) is not complete.

Hint:

  • Do not forget to prove that sequences fail to converge to ANY point.

2.3

Add two point \(-\infty\) and \(+\infty\) to \(\R\) and call \(\bar{\R}=\R\cup\{\pm \infty\}\).

Set \(\bar{d^*}(x,y)=d^*(x,y)\) if \(x,y\in \R\).

Specify \(\bar{d^*}\) on all other pairs of points of \(\bar{\R}\) so that that \(\bar{\R},\bar{d^*}\) becomes complete.

Make sure to prove that \(\bar{d^*}(x,y)\) is a distance and that \(\bar{\R}\) is complete!

3

Let \(X\) be a metric space. Recall the following definition: \(L\in X\) is a limit point of a sequence \((x_{n}\in X)_{n\in\N}\) if there exists a subsequence \(y_{n}\) of \(x_{n}\) such that \(y_{n}\to L\).

3.1

  • Let \(x_{n}\in\R\) be a real sequence. Show that \(\limsup_{n}x_{n}\) is a limit point of \(x_n\) (assuming that \(\limsup_{n}x_{n}\ne \pm \infty\)).
  • Let \(x_{n}\in\R\) be a real sequence. Show that \(\liminf_{n}x_{n}\) is a limit point of \(x_n\) (assuming that \(\liminf_{n}x_{n}\ne \pm \infty\)).

3.2 ​   hard

  • Let \(x_{n}\in\R\) be a real sequence. Show that \(\limsup_{n}x_{n}\) is the largest limit point of \(x_n\) (assuming that \(\limsup_{n}x_{n}\ne \pm \infty\)).

3.3

  • Let \(x_{n}\in\R\) be a real sequence such that \(\lim_n x_n=L\). Show that then \(\limsup_{n}x_{n}=\liminf_{n}x_{n}=L\).

3.4

  • Let \(x_{n}\in\R\) be a real sequence such that \(\limsup_{n}x_{n}=\liminf_{n}x_{n}\). Show that then \(\lim_{n}x_n\) exists and \(\lim_{n}x_n=\limsup_{n}x_{n}=\liminf_{n}x_{n}\). (This is the converse of the above implication.) Hint: Use the sandwich.

4

In this problem you should heavily use:

\begin{equation*} (1+z)^{n}\geq1+nz \qquad |z|\leq 1, n\in\N \end{equation*}

You can consider proving that

\begin{equation*} (1+z)^{n}\geq1+nz \qquad |z|\leq 1, n\in\Z \end{equation*}

or using the trick:

\begin{equation*} (1+z)^{n}=\frac{1}{(\frac{1}{1+z})^{n}}=\frac{1}{(1-\frac{z}{1+z})^{n}} \end{equation*}

and applying the inequality above (valid as long as \(\frac{z}{1+z}\) is in an appropriate range).

4.1

Show that \(\lim_{n}\big(1+\frac{1}{n}\big)^{n}\) and \(\lim_{n}\big(1+\frac{1}{n}\big)^{n+1}\) exist and are the same. We call this number \(e\).

Hint: Use monotonicity of the sequences proven in HW01.

4.2

Show that \(\lim_{n}\big(1+\frac{x}{n}\big)^{n}\) exists for each \(x\in\R\).

Hint: You may use the monotonicity of the sequence above, proven in HW01.

We use denote the limit above via a special function:

\begin{equation*} \mathrm{exp}(x):= \lim_{n}\big(1+\frac{x}{n}\big)^{n} \end{equation*}

4.3 ​   hard

Show that \(e^{x}e^{y}=e^{x+y}\) i.e. that

\begin{equation*} \lim_{n}\big(1+\frac{x}{n}\big)^{n} \lim_{n}\big(1+\frac{y}{n}\big)^{n}= \lim_{n}\big(1+\frac{x+y}{n}\big)^{n} \end{equation*}

Hint:

  • You know that the limits above exist so it is sufficient (by algebra of limits)

to show that

\begin{equation*} \lim_{n} \frac{\big(1+\frac{x}{n}\big)^{n} \big(1+\frac{y}{n}\big)^{n}}{\big(1+\frac{x+y}{n}\big)^{n}}=1 \end{equation*}

4.4

Let us denote \(\mathrm{exp}(x):= \lim_{n}\big(1+\frac{x}{n}\big)^{n}\)

Show that

\begin{equation*} \lim_{n\to \infty } \mathrm{exp}(\frac{1}{n})=1 \end{equation*}
  • Be careful when writing the solution: the order of taking limits is important! And generally limits do not commute.

Also, be careful what you use as \(n\) (name your indexes well).

  • You may NOT use that \(\mathrm{exp}(1/k)=e^{1/k}\) intended as the \(k\)-th root of the real number \(e\) UNLESS you deduce it from the previous point of this problem.

4.5 ​   hard optional

Show that

\begin{equation*} \lim_{n\to \infty } n(e^{1/n}-1)=1 \end{equation*}

4.6 ​   very_hard optional

Find

\begin{equation*} \lim_{n\to \infty } n^2(e^{1/n}-1-1/n) \end{equation*}