Real Analysis

Prove that the function defined by $f(x) = \frac{1}{1 + x^2}$ is uniformly continuous on $\mathbb{R}$.

Prove that the function $f(x)$ is continuous or discontinuous at $x = 2$ and $x = 3$ using the given piecewise function: $f(x) = \sqrt{x+2}$ when $x < 2$, $f(x) = x^2 - 2$ when $2 \leq x < 3$, $f(x) = 2x + 5$ when $x \geq 3$.

Determine if the function $f(x) = 2x + 5$ when $x < -1$, $f(x) = x^2 + 2$ when $x > -1$, and $f(x) = 5$ when $x = -1$ is continuous or discontinuous at $x = -1$. If discontinuous, identify the type of discontinuity.

Prove that the square root function is continuous at $x = 0$ using the epsilon-delta definition of continuity.

Prove that the square root function is continuous at all positive numbers using the epsilon-delta definition of continuity.

Consider the sequence $a_n = \frac{n}{2}$. Prove that this sequence diverges to positive infinity.

Prove that the sequence $a_n = -\frac{n}{2}$ diverges to negative infinity.

Prove that the sequence $a_n = (-1)^n$ diverges.

Consider the series with $\frac{1}{k!}$. The question is: 'Is this a convergent series?'

Consider a complicated looking series, but with a power involved. Use the root test to determine if this series is convergent.

Use the ratio test on $\frac{1}{e} + \frac{2}{e^2} + \frac{3}{e^3} + \ldots + \frac{n}{e^n} + \ldots$ to determine if the series converges or diverges.

What does the ratio test say about the convergence/divergence of the series $\sum_{n=1}^{\infty} \frac{(-1)^n}{(n+3)^4}$?

Prove that the set of real numbers between 0 and 1, where their decimal expansion only contains the digits 3 and 4, is uncountable using Cantor's diagonalization argument.

Prove Cantor's Theorem which states that for any set $A$, the power set of $A$ always has a cardinality strictly greater than $A$.

For the set of reciprocals of all natural numbers, find the greatest lower bound (infimum) and the least upper bound (supremum).

Given the set composed of terms rac{m}{m+n} for $m, n$ in natural numbers, find the greatest lower bound (infimum) and the least upper bound (supremum).

Prove that if $S$ is an upper bound for a set $A$, then $S$ is the least upper bound (supremum) of $A$ if and only if for all $epsilon > 0$, there exists an a  in A such that $S - epsilon < a$.

Take a subset of the rational numbers Q such that $x^2 < 2$ for all x in Q. Show that this set does not have a supremum in Q.

If a function $f$ is continuous at a point $x_0$, then the limit of the function $f$ as $x \to x_0$ exists and is equal to $f(x_0)$. Verify this for the function $f(x) = |x|$ at $x_0 = 0$.

Give an example of a function that is discontinuous everywhere.