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MATH063 real analysis课程简介
The study of metric spaces and real-valued functions is an important branch of mathematical analysis. In this field, mathematicians explore the properties of mathematical structures known as metric spaces, which consist of a set of points along with a distance function that assigns a distance between any two points in the set. These structures are used to define and study various types of functions, such as real-valued functions, which map points in a metric space to real numbers.
One of the central concepts in this field is the convergence of a sequence of points in a metric space. A sequence is said to converge to a limit if its distance from the limit point becomes arbitrarily small as the sequence progresses. This concept is used to define important properties of functions such as continuity and uniform continuity. A function is said to be continuous if the limit of the function at a given point is equal to the function’s value at that point. A function is uniformly continuous if the difference between the function’s values at two points becomes arbitrarily small as the distance between those points becomes arbitrarily small.
PREREQUISITES
Sequences and series of functions are also studied in this field. For example, a sequence of functions can be said to converge pointwise if the limit of the sequence at each point in the domain exists. Alternatively, the sequence can be said to converge uniformly if the difference between the function values becomes arbitrarily small uniformly across the domain.
Differentiation and Riemann-Stieltjes integration are two additional topics that are explored in this field. The derivative of a function measures the rate at which the function changes at a given point, while the Riemann-Stieltjes integral generalizes the Riemann integral to allow for integration with respect to a more general class of functions.
Overall, the study of metric spaces and real-valued functions is a rich and important field that has many applications in mathematics and other areas of science and engineering.
MATH063 real analysis HELP（EXAM HELP， ONLINE TUTOR）
Using the inequality $\frac{1}{k^2} \leq \frac{1}{k(k-1)}=\frac{1}{k-1}-\frac{1}{k}$, prove that the series $\sum_{k=1}^{\infty} \frac{1}{k^2}$ converges.
Suppose $a_k \geq 0$ for all $k$. Prove that if $\sum a_k$ converges, then
$$
\sum_{k=1}^{\infty} \frac{\sqrt{a_k}}{k} \text { converges. }
$$
If $\sum_{k=1}^{\infty} a_k$ and $\sum_{k=1}^{\infty} b_k$ both converge, prove each of the following:
a. $\sum_{k=1}^{\infty} c a_k$ converges for all $c \in \mathbb{R}$.
b. $\sum_{k=1}^{\infty}\left(a_k+b_k\right)$ converges.
Suppose $b_k \geq a_k \geq 0$ for all $k \in \mathbb{N}$.
a. If $\sum_{k=1}^{\infty} b_k$ converges, prove that $\sum_{k=1}^{\infty} a_k$ converges.
b. If $\sum_{k=1}^{\infty} a_k$ diverges, prove that $\sum_{k=1}^{\infty} b_k$ diverges.
Textbooks
• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.
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