# real analysis

1. ### I Finite expansion of a fraction of functions

I am having a problem finding the right order above and below to find the finite expansion of a fraction of usual functions assembled in complicated ways. For instance, a question asked to find the limit as x approaches 0 for the following function I know that to solve it we must first find...
2. ### I Do I need induction to prove that this sequence is monotonic?

I think the initial assumptions would allow me to prove this without induction. Suppose $(x_n)$ is a real sequence that is bounded above. Define $$y_n = \sup\{x_j | j \geq n\}.$$ Let $n \in \mathbb{N}$. Then for all $j \in \mathbb{N}$ such that $j \geq n + 1 > n$ $$x_{j} \leq y_n.$$...
3. ### I Rudin: theorem 1.21

Summary: Rudin theorem 1.21 He has said that as t=X/(X+1) then t^n<t<1 then maximum value of t is 1. then in the next part he has given that t^n<t<x. as maximum value of t is less than 1 why has he given that t<x ?
4. ### Help with a real analysis problem

I tried to prove this by absurd stating that there is no such $\mu'$ but i couldn't get anywhere...
5. ### A What type of function satisfy a type of growth condition?

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$. Is there any class of function and some type of "growth conditions" such that bounds like below can be established: \begin{equation} ||f(x)||\geq g\left( \text{dist}(x,\mathcal{X})\right), \end{equation} with $\mathcal{X}:= \{x:f(x)=0\}$ (zero...
6. ### Positive derivative implies growing function using Bolzano-Weierstrass

I'm stuck on a proof involving the Bolzano-Weierstrass theorem. Consider the following statement: $$f'(x)>0 \ \text{on} \ [a,b] \implies \forall x_1,x_2\in[a,b], \ f(x_1)<f(x_2) \ \text{for} \ x_1<x_2$$ i.e. a positive derivative over an interval implies that the function is growing over the...
7. ### Function Continuity Proof in Real Analysis

Homework Statement We've been given a set of hints to solve the problem below and I'm stuck on one of them Let f:[a,b]->R , prove, using the hints below, that if f is continuous and if f(a) < 0 < f(b), then there exists a c ∈ (a,b) such that f(c) = 0 Hint let set S = {x∈[a,b]:f(x)≤0} let c =...

30. ### Light in a cup (Can you explain this phenomenon?)

Can anyone explain the behavior of light I came across as I sat in my lounge this evening having a nice cup of Mocha . Hint ( I am sitting in a room with some led ceiling lights on) can you: 1.Guess how many Led lights I have on 2.Explain the appearance of light which is looking like a typical...