### Real analysis

Prove: subsequences of a convergent sequences converge to the same limit as the original sequence

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Prove: subsequences of a convergent sequences converge to the same limit as the original sequence

Prove that the sequence defined by X1=3 and X_n+1=1/(4-X_n) converges

Show that lim inf a_n = lim sup a_n if and only if lim a_n exists

Show that if x_n <= y_n <= z_n for all n belong to N and if lim x_n=limz_n=L then lim y_n=L as well

A) Show that if (b_n)-->b,then the sequence of absolute values Absolute value of b_n converges to absolute value of b

Show that limits, if exist, must be unique. In other words, assume lim an=L1 and lim an=L2 and prove that L1=L2

Let xn(smaller n)>=0 for all n belong to N a) if (xn)-->0, show that(sqrt[xn])-->0 b)if (xn)-->x,show that(sqrt[xn])-->x

Using the definition of convergence of a sequence show that the following sequences converge to the proposed limit: 1-lim 1/(6n^2+1)=0 2-lim 2/sqrt[n+3]=0 3-lim (3n+1)/(2n+5)=3/2

1- if A1,A2,A3,...,Am are each countable sets, then the union A1 U A2 U A3...U Am is countable 2- if An is a countable set for each n belong to N,then Un=1(to infinity) An is countable

Proof: if A subset or equal of B and B is countable, then A is either countable, finite or empty.

Prove that if a is an upper bound for A and if a is also an element of A, then it must be that a=sup A

Let A subset or equal of R be bounded above and let c belong to R.Define the sets c+A and cA by c+A={c+a : a belong to A} and cA={ca : a belong to A}. 1-show that sup(c+A)=c+ Sup A 2-if c>=0,show that sup(cA)=cSupA 3-postulate a similar type of statement for sup(cA)for the case c<0.

Assume that A And B are nonempty, bounded above and satisfy B subset or equal of A. Show that sup B<= sup A

Let y1=1,and for each n belong to N define y_n+1=(3y_n+4)/4. a-Use induction to prove that the sequence satisfies y_n<4for all n belong to N. b-use another induction argument to show the sequence(y1,y2,y3,...)is increasing.

Given a function f and a subset A of its domain, let f(A) represent the range of f over the set A; f(A)={f(x) : x belong to A}. a-Let f(x)=x^2. If A=[0,2](the closed interval{x belong to R : 0<=x<=2}) and B=[1,4],find f(A) and f(B).Does f(A intersection B)=f(A) intersection f(B) in this case?.Does f(A U B)=f(A) U f(B)?. b-Fi

Please see the attached file for the fully formatted problems. Suppose that is not a perfect nth power, i.e K is not equal to (a) Prove that is not a member of Q, the set of all rational numbers. (b) Infer that the nth root of a natural number is either a natural number or it is irrational.

Antiderivative of: (t5 + 6t3) dt

Show that the sum from 0 to infinity of (1-x)x^n does not converge uniformly on [0,1]. What subintervals of [0,1] does it converge uniforlmly on?

What is the fifth term of the series an = n + an-1' if a0 = -3.

Find an article through newspapers, magazines, professional journals, etc and find at least two examples of data that are best modeled using linear formulae. Describe the importance of each example and why a linear model is appropriate for the data. Note that we are referring to a linear model not simply a time chart where dots

Evaluate Limit[tan (a*theta)/sin(b*theta)] as theta approaches zero.

Calculate limit (1/x^2) as x approaches infinity

Prove that Lim (3x)=6 as x approaches 2 using Epsilon-Delta definition.

Determine whether the series Sum(n!/n^n) n=1..infinity is absolutely convergent, conditionally convergent or divergent.

Test the series (in the attached file) for convergence or divergence by using the Comparison Test or the Limit Comparison Test.

Please see the attached file for the fully formatted problem. What is the maximum of F = x1 +x2 +x3 +x4 on the intersection of x21 +x22 +x23 + x24 = 1 and x31+ x32+ x33+ x34= 0? As this is an analysis question, please be sure to be rigorous and as detailed as possible.

Verify Fubini's Theorem for an integral evaluated over an equilateral triangle. (My notes from class-make a function up, similar to something used in question 1, but change powers of x & y.) You should discuss fully the reasons for the limits of integration in your solution.

Let f(x) be integrable on [a,b], and let g(x) be nondecreasing and continuously differentiable on [a,b]. Let {p be element of P} be a partition of [a,b], and define U(f,g,p) = SIGMA (Mi(g(the ith term of x) - g(the (i-1)th term of x))) as i=1 to n L(f,g,p) = SIGMA (Ni(g(the ith term of x)-g(the (i-1)th term of x))) as i=1 t

Let f:[a,b] mapped to the Reals be a function that is integrable over [a,b], and let g:[a,b] mapped to the Reals be a function that agrees with f except at two points. Prove g is integrable over [a,b].

The problem is to determine the radius of convergence of the Taylor Series for each of the functions below centered at x. We are to explain our conclusion in each case. I would like to see how to work each problem (including what the Taylor Series is) and what the explanation is. a) centered at and NOTE: I know the