# Sergei Yakovenko's blog: on Math and Teaching

## Wednesday, December 28, 2011

### תרגיל בית מס’ 3

אין זמן למנוחה, ועל כן מוגש בפניכם תרגיל בית מס’ 3.

שימו לב כי יש לכם את כל הכלים לפתור את התרגיל, למרות שהוא עלול להיראות קשה ממבט ראשון.

תאריך אחרון להגשה: 10 בינואר.

בהצלחה!

## Continuity and limits of functions of real variable

In the first lecture we introduce the notion of continuity of a function at a given point in its domain and a very close notion of a limit at a point outside of the “natural” domain.

This notion is closely related to the notion of sequential limit as introduced earlier. This paves a way to generalize immediately all arithmetic and order results from numeric sequences to functions.

The novel features involve one-sided limits, limits “at infinity” and continuity of composition of functions.

The (unfinished) notes, to be eventually replaced by a more polished text, are available here: follow the updates, this temporary link will eventually be erased.

## Limits of sequences

We spend some time considering different flavors of “limit behavior”: stabilization, approximate stabilization etc. A sequence is called converging, if it $\varepsilon$-stabilizes for any positive accuracy $\varepsilon>0$.

To show that passing to a limit “respects” arithmetic operations, we need to work out a bit of “interval arithmetic” with a special attention to the division which may cause problems.

We discuss the weaker notion of a partial limit (accumulation point) and study under what assumptions a unique partial limit is the genuine limit.

Finally, we show that monotone bounded sequences always converge. This is one of the most powerful tools to show that the limit exists when it is not possible to compute it explicitly.

## Tuesday, December 27, 2011

### סיכום תירגולים בנושא סדרות והתכנסות

שלום לכולם,

מצורף קישור לקובץ בו תמצאו סיכום של הנושאים שלמדנו בשבועות האחרונים.  כל הערה/תיקון יתקבלו בברכה רבה.

דימה

## Tuesday, December 13, 2011

### קובץ תרגילים לא להגשה

בקישור זה תוכלו למצוא שני תרגילים פשוטים בנושא שברים משובלים.
לא להגשה.

דימה

### תרגיל בית מס’ 2

שלום לכולם,
קובץ תרגילים מס’ 2 נמצא .
בקישור הזה

להגשה עד 22 בדצמבר.

דימה

## Construction of real numbers

The idea of extending the number system from the set of rational numbers $\mathbb Q$ by adjoining roots of polynomial equations is very interesting, but faces obvious difficulties: we need to treat all possible polynomial equations, and this still give us no guarantees whatsoever that transcendental equations (trigonometric, exponential etc). will be solvable when we expect them to be.

The alternative is to extend the set of rationals by adding “solutions to systems of inequalities”. In order for such a system to represent a unique “new” number, the equations need to be consistent (compatible between themselves) and possess some uniqueness property.

These two requirements can be implemented by consideration of the so called Dedekind cuts, which can be informally considered as sets of rational “approximations” (lower and upper) for the missing number.

In the lectures we pursue this strategy and explain how the cuts can be compared, how arithmetic operations on the cuts can be defined and why the addition of all possible cuts results in a “complete” number system.

The detailed exposition, as before, is downloadable as a pdf file. Please take a time to signal (in the comments to this post or by any other way) about all errors, inevitable in the first draft.

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