Real Analysis

Consider the sequence $\{f_n\}$ defined by $f_n(x) = \frac{nx}{ 1 + nx}$ , for $x ≥ 0$. a) Find $f(x) = \lim _{n→∞ }f_n(x).$ b) Show that for $a > 0$, $\{f_n\}$ converges uniformly to $f$ on $[a,∞)...

If $T$ is a mapping from a complete metric space $(M, d)$ into itself such that $T^m$ is a contraction mapping for some $m \in \mathbb{N}$, then show that $T$ has a unique fixed point.

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfy |f(x)-f(y)|\leq |x-y|^2,    \forall   x,y\in \mathbb{R}. Prove that $f$ is constanct.

Suppose that $\mu$ and $\nu$ are finite measures on a measurable space $(X,\mathcal{M})$. Prove that either $\nu \perp \mu$, or there exists $\epsilon > 0$ and a measurable set $E \subseteq X$ such...

Consider the mapping $F: \R^4\rightarrow\R^2,$ $(x_1,y_1,x_2,y_2)\mapsto(F_1(x_1,y_1,x_2,y_2),F_2(x_1,y_1,x_2,y_2))$ where $F_1:=x_1y_1+x_2y_2$ and $F_2:=x_1y_2-x_2y_1$.(a) Find the points in $\R^4$ w...

Prove that a closed subset of a compact set is compact. Please included details of the proof. 

Let $\varphi:\R\rightarrow\R$ be a $C^\infty$-smooth function with compact support. Prove that the following limit exists, and compute the limit.\lim_{\varepsilon\rightarrow0+} \int_{-\infty}^{\infty}...

Given topological spaces $X$ and $Y$, define  Let $X$ be a compact topological space and $V$ a normed vector space. Prove that $C(X,V)$ becomes a normed vector space if we define addition and scalar m...

Please see the attachement. 

Suppose that $\mu,\nu, \lambda$ are $\sigma$-finite measures on a measurable space $(X,\mathcal{M})$ and that Prove that  

let f: R->R be a continuous function such that f(0)=f(2)=1. then there exists c>0 such that f(c)=cprove this. 

Assume $\{f_n\}\subset L^+$, $f_n\rightarrow f$ pointwise, and $\int f=\lim \int f_n < \infty$. Then $\int_E f= \lim \int_E f_n$ for all $E \in M $. However show that this may not be true if $\int...

Suppose that $\mu$ and $\nu$ are finite measures on a measurable space $(X,\mathcal{M})$. Prove that either $\nu \perp \mu$ or there exists $\epsilon > 0$ and a measurable set $E \subseteq X$ such...