Compactness argument

Let $f : [0,1] \rightarrow \mathbb {R}$ be continuous and $k \in \mathbb {N}$. Prove that there is a real polynomial $P(x)$ of degree $\leq k$ which minimizes (for all such polynomials)

Let $(a_{n})$ be a sequence of nonzero real numbers. Prove that the sequence of functions $f_{n} : \mathbb {R} \rightarrow \mathbb {R}$

has a subsequence converging to a continuous function.

Let $f$ be a continuous function from the ball $B_n = \left\{ x \in \mathbb {R}^{n} \mid \left\| x\right\| < 1\right\}$ into itself. (Here, $\left\| \cdot \right\|$ denotes the Euclidean norm.) Assume $\left\| f(x)\right\| < \left\| x\right\|$ for all nonzero $x \in B_n$. Let $x_0$ be a nonzero point of $B_n$, and define the sequence $(x_k)$ by setting $x_k = f(x_{k-1})$. Prove that $\lim x_k = 0$.

Let $f : \mathbb {R}^{n} \rightarrow \mathbb {R}^{n}$ be continuously differentiable. Assume the Jacobian matrix $(\partial f_i / \partial x_j)$ has rank $n$ everywhere. Suppose $f$ is proper; that is, $f^{-1}(K)$ is compact whenever $K$ is compact. Prove $f(\mathbb {R}^{n}) = \mathbb {R}^{n}$.

Suppose $f$ is a continuously differentiable map of $\mathbb {R}^{2}$ into $\mathbb {R}^{2}$. Assume that $f$ has only finitely many singular points, and that for each positive number $M$, the set $\left\{ z \in \mathbb {R}^{2} \mid \left|f(z)\right| \leq M\right\}$ is bounded. Prove that $f$ maps $\mathbb {R}^{2}$ onto $\mathbb {R}^{2}$.

Let $S \subset \mathbb {R}^{3}$ denote the ellipsoidal surface defined by

Let $T \subset \mathbb {R}^{3}$ be the surface defined by

Prove that there exist points $p \in S$, $q \in T$, such that the line $\overline{pq}$ is perpendicular to $S$ at $p$ and to $T$ at $q$.

Let $X \subset \mathbb {R}^{n}$ be a closed set and $r$ a fixed positive real number. Let $Y = \left\{ y \in \mathbb {R}^{n} \mid \left|x-y\right| = r \text{ for some } x \in X\right\}$. Show that $Y$ is closed.

Let $A \subset \mathbb {R}^{n}$ be compact, $x \in A$; let $(x_i)$ be a sequence in $A$ such that every convergent subsequence of $(x_i)$ converges to $x$.

Let $X$ be a compact metric space and $f : X \to X$ an isometry. Show that $f(X) = X$.

Let $K$ be a nonempty compact set in a metric space with distance function $d$. Suppose that $\varphi : K \to K$ satisfies

for all $x \neq y$ in $K$. Show there exists precisely one point $x \in K$ such that $x = \varphi (x)$.

Let $X$ be a reflexive Banach space. Prove that if the sequence $\left\{ x_{n}: n \geq 1\right\} \subset X$ is bounded in norm, then there exist a subsequence $\left\{ x_{n_{k}}: k \geq 1\right\}$ and an element $x \in X$ such that $x_{n_{k}} \xrightarrow {w} x$.

Prove that if a function is lower semicontinuous on the segment $\left[a, b\right]$ and takes only finite values, then it attains its least value on $\left[a, b\right]$ (and hence is bounded below). Is it true that it is bounded above?

Let

Prove that the image $P(F)$ of any closed set $F \subset \mathbb {C}$ is again a closed set. Is this true for an arbitrary polynomial in two variables?

Let $\left\{ f_{n}\right\}$ be an equicontinuous, uniformly bounded sequence of realvalued functions on $\mathbb {R}$. Prove that there is a subsequence $\left\{ f_{n_{k}}\right\}$ that converges pointwise on $\mathbb {R}$ to a continuous function $f$ on $\mathbb {R}$ and that the convergence is uniform on bounded subsets of $\mathbb {R}$.

Let $g: \mathbb {R} \rightarrow [0,+\infty )$ and suppose that, for $f \in C^{1}[0,1]$,

Finally, let $\mathcal{F}=\left\{ f \in C^{1}[0,1]:\left\| f\right\| \leq 1\right\}$. Prove that there is a function $h \in \mathcal{F}$ that minimizes $I(f)$, i.e., $I(h)=\inf \left\{ I(f): f \in \mathcal{F}\right\}$.

If a monotone sequence of continuous functions converges on a closed interval to a continuous function it converges uniformly. (Theorem of Dini.)

The analytic functions $f_{1}(z),f_{2}(z),\ldots ,f_{n}(z)$ are regular and single-valued in the connected closed domain $\mathfrak {D}$; let $c_{1},c_{2},\ldots ,c_{n}$ denote constants. If the function

does not vanish identically the number of its zeros in $\mathfrak {D}$ cannot exceed a certain upper bound which depends on $f_{1}(z),f_{2}(z),\ldots ,f_{n}(z)$ and $\mathfrak {D}$ but does not depend on $c_{1},c_{2},\ldots ,c_{n}$. (Problem 206.2 is less general but more precise.)

Let $\left\{ f_{n}\right\}$ be an equicontinuous, uniformly bounded sequence of continuous real-valued functions on $\mathbb {R}$. Show that there is a subsequence of $\left\{ f_{n}\right\}$ that converges pointwise on $\mathbb {R}$ to a continuous function on $\mathbb {R}$ and that the convergence is uniform on each bounded subset of $\mathbb {R}$.

Let $S$ be a countable set and $\left\{ f_{n}\right\}$ a sequence of real-valued functions on $S$ that is pointwise bounded on $S$. Show that there is a subsequence of $\left\{ f_{n}\right\}$ that converges pointwise on $S$ to a real-valued function.

In Euclidean space $\mathbb {R}^{n}$ consider the closed unit ball $B = \left\{ x \in \mathbb {R}^{n} \mid \left\| x\right\| \leq 1\right\}$. Let $f$ map $B$ into $B$ and be Lipschitz with Lipschitz constant 1. Without using the Brouwer Fixed-Point Theorem, show that $f$ has a fixed-point.