Real analysis problem

I need an expert who can do the following real analysis problem. Don't need details. Just some hints.

I prefer if u give hints to solve the first question in pm so that I can understand ur expertise.

All R stands for real numbers.

Suppose that the function f:[a,b]->R is bounded and that it is continuous except at one point x_0 in the open interval (a,b). Prove that f:[a,b]->R is integrable.

Suppose that the functions f:[a,b]->R and g:[a,b]->R are continuous. Prove that integral from a to b of |f+g| <= integral from a to b |f| + integral from a to b |g|.

The monotonicity property of the integral implies that if the functions g:[0,infinity)->R and h:[0,infinity)->R are continuous and g(x)<=h(x) for all x>=0, then the integral from 0 to x of g <= integral from 0 to x of h for all x>=0. Use this and the First Fundamental Theorem (integrating derivatives) to show that each of the following inequalities implies its successor:

cos(x)<=1 if x>=0

sin(x)<=1 if x>=0

1-cos(x)<=(x^2)/2 if x>=0

x-sin(x)<=(x^3)/6 if x>=0


x-(x^3)/6<=sin(x)<=x if x>=0.

Under the assumptions of the Alternating Series Test, define s=sum from k=1 to infinity of [(-1)^(k+1)]*a_k. Prove that for every index n, |s-[sum from k=1 to n of [(-1)^(k+1)]*a_k]|<=a_(n+1).

Use the Cauchy Convergence Criterion for Series to provide another proof of the Alternating Series Test.

Determine whether the sequences, f_n(x)=e^(-nx^2) for all x, f_n(x)= 1 if x=k/2^n for some natural number k or 0 otherwise, and f_n:[0,1]->R such that f_n(0)=f(2/n)=f_n(1/n)=n, and f_n is linear on the intervals [0,1/n], [1/n,2/n], and [2/n,0], converge uniformly.

Suppose that the sequences {f_n:D->R} and {g_n:D->R} converge uniformly to the functions f:D->R and g:D->R, respectively. For any two numbers alpha and Beta, prove that the sequence {alpha*f_n + beta*g_n: D->R} converges uniformly to the function alpha*f + beta*g: D->R.

For each natural number n, let the function f_n:R->R be bounded. Suppose that the sequence {f_n} converges uniformly to f on R. Prove that the limit function f:R->R also is bounded.

Навыки: Математика

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ID проекта: #1615059

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I can help you.

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Hello ! I can help you . I have master degree at analysis . I send you hints to solve first question see your Inbox

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