## Show that \$[2,infty)\$ is also uncountable

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 …

## Axiom of Completeness to prove intermediate value theorem

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 …

## \$f(x)={1over (x-1)(x+2)}\$ Taylor series.

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 …

## Prove this function is the identity

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 …

## Proof that a sequence of set has a set dense somewhere in \$[a,b]\$

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]\$ …

## Why if \$T\$ is not continuous, then for each \$nin N\$, there exists \$x_nin X\$ such that \$||Tx_n||ge n||x_n||\$

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 help using the limit comparison test for \$sum frac{1}{sqrt{n^2 + 1}}\$

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 …

## Given \$f(x)=big(frac{1}{x}big)^{1/3}\$. Is the area bounded by the function and the \$x\$ axis finite?

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 \$…

## Test whether the follwing sets are compact or NOT?

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) …

## Norm of an integral operator on the space of continuous functions

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 …