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\title{ Math 5210,  HW I \\ due Friday, Jan 31 }
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1)  In this and the following problem, use $d(x,y)=\max( |x_1-y_1|, |x_2-y_2|)$ as the distance function on $\mathbb R^2$. 
Use $\epsilon$-$\delta$ definition of continuity to prove that the multiplication map $\mathbb R^2 \rightarrow  \mathbb R $ is continuous. 
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2) Let $p_i : \mathbb R^2 \rightarrow \mathbb R$ be the projection on the $i$-th coordinate. Prove that $p_i$ is continuous. 
Let $(X,d)$ be a metric space. Let $f: X \rightarrow \mathbb R^2$ be a map, and write $f(x)=(f_1(x), f_2(x))$ for every $x\in X$. 
In particular we have two functions $f_i: X\rightarrow \mathbb R$, $i=1,2$. Prove that $f$ is continuous if and only if $f_1$ and $f_2$ are. 

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3) Let $f: \mathbb R \rightarrow \mathbb R$ given by $f(x)=x^n$. Use the inductive definition $x^n=x\cdot x^{n-1}$ and previous exercises to prove that 
$f$ is continuous. 

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4) Let $f:[a,b] \rightarrow \mathbb R$ be a continuous function such that $f(x) \geq 0$ for all $x\in [a,b]$.  Prove that 
\[ 
\int_a^b f = 0 
\] 
implies $f(x)=0$ for all $x\in [a,b]$. 

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5) Let $(X,d)$ be a metric space. Let $(x_n)$ and $(y_n)$ be two Cauchy sequences in $X$. Prove that $(d(x_n,y_n))$ is a Cauchy sequence in $\mathbb R$. 

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6)  Let $K\subset \mathbb R$ be a set consisting of $0$ and all $1/n$, $n=1,2,3,\ldots$. Prove that $K$ is compact directly using the definition, i.e. every open cover has a 
finite subcover. 

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7)  Let $F_1 \supseteq F_2 \supseteq  \ldots $ be a descending sequence of non-empty compact subsets. Prove that $\cap_{n=1}^{\infty} F_n$ is non-empty. 

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8) Let $(X,d)$ be a metric space and $f_n$ a sequence of continuous functions  $f_n : X \rightarrow \mathbb R$ uniformly converging to $f$. Let $x_n$ be a 
sequence of points in $X$ such that $\lim_n x_n=x\in X$. Prove that $\lim_n f_n(x_n)=f(x)$ . 

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9) A subset $\mathbb R^n$ is convex if for any two points $x,y\in C$, the segment $[x,y]$ is contained in $C$. Prove that $C$ is connected. 


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