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# 数学代写|MATH5201 Real Analysis

## MATH5201课程简介

Basic metric topology, sequences and series, continuous functions, differentiable functions of one variable, Riemann integration, uniform convergence, Fourier series.
Prereq: Grad standing, or permission of department. Not open to students with credit for 651.

Credit Hours

5.0

## Prerequisites

The theorems of real analysis rely on the properties of the real number system, which must be established. The real number system consists of an uncountable set R ,together with two binary operations denoted + and ⋅, and an order denoted <. The operations make the real numbers a field, and, along with the order, an ordered field.

## MATH5201 Real Analysis HELP（EXAM HELP， ONLINE TUTOR）

The sequence defined recursively by
$$f_1=1, f_2=1, \quad f_{n+2}=f_n+f_{n+1}$$
is called the Fibonacci sequence. It is possible to find an explicit formula for this sequence. Give it a try.

Show that, once it is known that the interval $(0,1)$ cannot be expressed as the range of some sequence, it follows that any interval $(a, b),[a, b),(a, b]$, or $[a, b]$ has the same property.

A set (any set of objects) is said to be countable if it is either finite or there is an enumeration (list) of the set. Show that the following properties hold for arbitrary countable sets:
(a) All subsets of countable sets are countable.
(b) Any union of a pair of countable sets is countable.
(c) All finite sets are countable.

Show that the following property holds for countable sets: If
$$S_1, S_2, S_3, \ldots$$
is a sequence of countable sets of real numbers, then the set $S$ formed by taking all elements that belong to at least one of the sets $S_i$ is also a countable set.

Define a relation on the family of subsets of $\mathbb{R}$ as follows. Say that $A \sim B$, where $A$ and $B$ are subsets of $\mathbb{R}$, if there is a function
$$f: A \rightarrow B$$
that is one-to-one and onto. (If $A \sim B$ we would say that $A$ and $B$ are “cardinally equivalent.”) Show that this is an equivalence relation, that is, show that
(a) $A \sim A$ for any set $A$.
(b) If $A \sim B$ then $B \sim A$.
(c) If $A \sim B$ and $B \sim C$ then $A \sim C$.

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