Foundation of analysis questions.

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#	Question
1	If $f:mathbb{R} ightarrow mathbb{R}$ is given by $f(x)=3x+7$ , prove that it is one to one. Mathematical Calculation
2	Let A be a set of non-zero integers and let $equiv$ be the relation on AxA defined by (a,b) $equiv$ (c,d) wherever ad=bc. Prove that $equiv$ is an equivalence relation. Mathematical Calculation
3	Use de Moivre's theorem to express $cos4 heta$ and $sin4 heta$ in terms of powers of $cos4 heta$ and $sin4 heta$ . Mathematical Calculation
4	Find the roots of $z^3-z^2 -z-2=0$ , given that z=2 is a root. Mathematical Calculation
5	Determine if the following sets are bounded or not bounded. $S={{x in mathbb{R}\|0leqslant x<1}}$ $S={{n\|ninmathbb{N}}}$ $S={{frac{1}{n}\|ninmathbb{N}}}$ . Prove that if the least upper bound of a subset $S$ of $Re$ exists, then it is unique. Mathematical Calculation
6	Define what it means for a sequence of real numbers $(x_n)_{ n=0}^{alpha}$ to converge to zero. Mathematical Calculation
7	Using the definition prove that both of the following sequences converge to zero. $x_n ={frac{1}{2n}}$ $x_n ={frac{1}{sqrt{n}}}$ Mathematical Calculation
8	Suppose $a=1$ .Show $P$ is true for all $ngeq 1$ where $P$ is defined as follows. $P(n):1+a+a^2+ ... +a^n=frac{a^{n+1}-1}{a-1}.$ Mathematical Calculation