MATH 3150 Midterm 1 Review

notereal-analysis

Exam Format

Problem 1

1. Compute $\sup ,\inf$ of a set.
2. Work with $\sup$, $\inf$ for some set.

Definition: $M=\sup S$ if

• $M$ is an upper bound of $S$.
• If $M^{\prime }$ is another upper bound, then $M^{\prime }\ge M$.

Property: $M=\sup S$ if and only if,

• $M$ upper bound of $S$
• $\forall \epsilon >0$, $\exists x\in S$ such that $x\le M\le x+\epsilon$.

If $(x_n{)}_n$ is monotonically increasing, then $\sup \left\{x_n:n\right\}={\lim }_{n\to \infty }(x_n)$

Problem 2

1. Use $\epsilon$ definition to calculate the limit of a sequence.
2. Apply the $\epsilon$ definition to prove properties of sequences (may need to use the triangle inequality).

Problem 3

Apply monotone convergence theorem for a sequence given recursively.

1. Bounded (needs induction)
2. Monotone (needs induction)
3. Limit

$(x_n{)}_n$ monotonically increasing implies $x_n\ge x_1$ (lower bound), so only need upper bound.

Not Tested

1. Set operations ($\cup ,\cap ,$ DeMorgan’s law)
2. Simple induction (like $1+2+\cdots +n=\frac{n(n+1)}{2}$)
3. Subsequence-related problems (will be on midterm 2).

Proving a Sequence is Monotone

$a_{n+1}=f(a_n)$.

1. Induction If $a_{n-1}\le a_n$ prove $a_n\le a_{n+1}$. We have $a_n\le a_{n+1}⟺f(a_{n-1})\le f(a_n)$.
2. Check directly $a_{n+1}\ge a_n⟺f(a_n)\ge a_n$ and solve in $a_n$.

Common Mistakes

1. A sequence being convergent does NOT necessarily imply that the sequence is monotone.