Question: Assume $A$ is the only uncountable set that we currently know of, where $A= { xin (0,1): x text{ is a decimal fraction consisting only of combinations of 0 and 1} }$ From …

# Category: Mathematics Real Analysis

Mathematics Real Analysis Questions

I am having a little trouble understanding one of the steps in this proof. From Stephen Abbott’s Analysis: Using AoC to prove the IVT: TO simplify matters, consider $f$ as a continuous function …

Find Taylor series around $x_0=0$ for: $$f(x)={1over (x-1)(x+2)}=(text{By a hintby simple algebra}){1over 3}left[{1over x-1}-{1over x+2}right]$$. Check where the series converges to the …

I’m trying to prove if $f:mathbb Rto mathbb R$ with the properties $f(x+y)=f(x)+f(y)$, $f(xy)=f(x)f(y)$ for any $x,yin mathbb R$ and $f(1)=1$ is $1-1$. I’m solving this question proving $f$ is …

Is the following proof correct? Proposition: if we have a sequence of set $U_i$ such as $bigcup_{iin mathbb{N}} U_i=[a,b]$ then there exist a $i$ such as $U_i$ is dense somewhere in $[a,b]$ …

If $X$ $Y$ are Banach space, $T$ is a linear operator between them. I don’t understand the following statement: If $T$ is not continuous, then for each $nin N$, there exists $x_nin X$ such that $$|…

I need to determine whether the following series converges or diverges: $$sum_{n=1}^{infty} frac{1}{sqrt{n^2 + 1}}$$ I’m having trouble finding a series to compare this to but I was thinking …

Consider the function $f(x)=Big(frac{1}{x}Big)^{1/3}$ with $xin[-1,1]$. I want to find out wether the area bounded by the function and $x$ axis is finite? Using simple strategy (i.e integrating $…

I am trying to find which of the following sets are compact and possibly a valid reason for why so I can better understand the concept. Here are the following sets I am trying to find: 1) ${(x,y) …

The following is a question I came up with when I was studying the same problem in dimension 1 (for which also I have the questions that follows) but I put in generality. Let $U_1, U_2 subset …