Recent content by shortydeb

If a set of numbers contains "more elements" than a countable infinite set, that doesn't necessarily mean the set is uncountable. Look at the set of all integers and the set of all even integers. The set of all integers has more elements than the set of all even integers. The set of all even...

the axioms for a metric space state that for any two points in the metric space, their distance is a real (and finite) number.

pretty much, yes. can you determine what the matrix is?

https://en.wikipedia.org/wiki/Interior_point that page has a picture which might help.

Set (14n + 11)/(16n + 19) equal to 139/159 and see what you get for n.

You have to find a positive integer such that if you plug it into (14n + 11)/(16n + 19) you get a number greater than 139/159. 

There are two possibilities for x: either x is greater than or equal to 3, or x is less than 3. If x ≥ 3, what does the inequality look like (i.e. without the absolute value)?

yes, but you also have to substitute 2 for x in the denominator-- x itself is a function of x.

x^3 isn't concave downward on the entire real line. Your professor is saying that functions that are concave downward at every point do have their tangent lines above the graph.

Suppose x is a complex number, and not in f(C). Let M > |x|. Then there exists N such that if |z| > N, then |f(z)| > M. The set {y: |y| ≤ N} is compact. Edit: x is either a limit point of f(C), or it isn't. If it's not, choose M so that the open ball with radius M centered at (0,0) contains the...

you're not summing two divergent limits. You're summing two functions whose individual limits at the same point are both divergent.

Here's an idea for the first part: Split (1/2, 1) into a countable number of subintervals. Specifically, (1/2, 3/4), (3/4, 7/8), (7/8, 15/16), etc. Let the functional value for all the irrationals in (1/2, 1) be zero. Let the functional value for the rationals in each subinterval be (1/n)...

If a set has no limit points, then by definition it is closed.

"The distance" never actually becomes zero. Are you familiar with the epsilon-delta definition of the limit? To truly understand the concept of a limit, you need to understand that definition.

Technically you do use the chain rule, because 17y is a function of y.