Real analysis: test 1

Grading

This is a test for Real analysis: course.

1. Give two non-Euclidean metrics on $\mathbf{R}^{2}.$ Prove.
2. Prove that an open ball in a metric space is an open set.
3. Prove that a compact set in a metric space is bounded and closed.
4. Suppose $S,T$ are metric spaces and $f,g:S\rightarrow T$ are continuous functions. Prove that the set $A=\{x\in S:f(x)=g(x)\}$ is closed in $S.$ What can you say about $B=\{x\in S:f(x)\neq g(x)\}?$
5. State and prove the fundamental lemma of differentiation for $f:\mathbf{R}^{2}\rightarrow\mathbf{R}$.
6. State the definition of a differentiable function $f:\mathbf{R}^{N}\rightarrow\mathbf{R.}$ Give an example of a function $f:\mathbf{R}^{2}\rightarrow\mathbf{R}$ such that both partial derivatives of $f$ exist at $x=a,$ but $f$ is not differentiable.

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1. Use the Sandwich Theorem to prove that $\lim\limits_{x\rightarrow0}x^{n}=0$ for every positive integer $n.$
2. State the definition of the one-sided limit and state the basic theorems about it.
3. From the definition, show that $f(x)=x^{2}-1$ is uniformly continuous on $[0,1].$
4. State the Nested Intervals Theorem and provide examples that show that the conditions of the theorem cannot be removed.
5. State and prove the theorem about boundedness of continuous functions.
6. Prove that any subsequence of a convergent sequence converges.