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Real Analysis

Assuming that ff and gg are continuous at some point cc in their domain AA, prove that: 1. Kf(x)K \, \cdot \, f(x) is continuous at cc for any real number KK, 2. f(x)+g(x)f(x) + g(x) is continuous at cc, 3. f(x)g(x)f(x) \, \cdot \, g(x) is continuous at cc, 4. f(x)g(x)\displaystyle \frac{f(x)}{g(x)} is continuous at cc given that the denominator g(x)g(x) is non-zero on the entire domain.

Given the function f(x)=xf(x) = x for all x[0,1)x \in [0,1), prove that there is no point in the interval where f(x)=1f(x) = 1, hence the supremum is not attained.

Given the function f(x)=11+x2f(x) = \frac{1}{1 + x^2} for all , demonstrate that the supremum is attained at x=0x = 0, but the infimum is not attained at any point in R\mathbb{R}.

If a function ff is continuous in the closed interval [a,b][a, b], then it is bounded in [a,b][a, b].

Find the critical point, any local maximums or minimum, and the intervals where the function is increasing and decreasing for y=2x28x+6y = 2x^2 - 8x + 6.

Find the intervals where the function y=2x3+3x212x+8y = 2x^3 + 3x^2 - 12x + 8 is increasing and decreasing, and find any maximum or minimums, and find the critical points.

Find the critical numbers for a given function f(x)f(x), and determine whether any relative extrema occur at those points.

Prove that the Dirichlet function, defined on the interval [0,1][0, 1] as f(x)=1f(x) = 1 if xx is rational and f(x)=0f(x) = 0 if xx is irrational, is not Riemann integrable.

Show that the function f(x)=xf(x) = x is Riemann integrable on the interval [0,1][0, 1] and find its integral value.

If x is an even number, then x2x^2 is an even number.

Calculate the radius of convergence, find the interval of convergence, and determine the set of points at which the power series is convergent.

Find the radius and interval of convergence for the power series n=0xn3n\sum_{n=0}^{\infty} \frac{x^n}{3^n}.

Find the radius and interval of convergence for the power series n=1n4n(x3)2n\sum_{n=1}^{\infty} \frac{n}{4^n} (x-3)^{2n}.

Prove that the sequence 1n\frac{1}{n} approaches 0 as nn approaches infinity.

Prove the order limit theorem for convergent sequences, including the three statements:

1) If every term of a convergent sequence is at least 0, then the limit of that sequence is also at least 0.

2) If every term of one convergent sequence is less than or equal to some other convergent sequence, then their limits have that same relation.

3) If a convergent sequence is bounded below by some real number, then the limit of that convergent sequence is at least as big as that lower bound. Similarly, if a convergent sequence is bounded above by some real number, then its limit is less than or equal to that real number.

Show that the sequence an=2+1n+1a_n = 2 + \frac{1}{n+1} converges to a limit a=2a = 2.

Using Cauchy's criterion of convergence, examine the convergence of sequence fn=1+11!+12!++1n!f_n = 1 + \frac{1}{1!} + \frac{1}{2!} + \ldots + \frac{1}{n!}.

Using Cauchy's criterion of convergence, examine the convergence of sequence fn=11!12!+13!+(1)n+11n!f_n = \frac{1}{1!} - \frac{1}{2!} + \frac{1}{3!} - \ldots + (-1)^{n+1} \frac{1}{n!}, also find the limit.

Given a sequence (an)(a_n) and its subsequence (ank)(a_{n_k}), prove that the kth term of the subsequence is at least k terms along in the original sequence, i.e., nkkn_k \geq k.