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### Problem 1

Consider the function on $[0,\infty)$ given by $\psi(t) = \dfrac{t}{1+t}$.
1. $\psi$ is concave and increasing
2. $\psi$ is continuous at $0$
3. For any $\varepsilon > 0$ there exists $\delta > 0$ such that $\psi(t) < \delta$ implies $0\le t < \varepsilon$

### Problem 2

Let $d_j$ be a sequence of quasimetrics on $X$ that separates points, and $d$ the metric constructed from the $d_j$ as in class. Let, for each positive integer $N$, $x\in X$ and $\varepsilon>0$,
$B_N(x,\varepsilon) = \{ y\in X: d_j(x,y) < \varepsilon, j = 1, 2, \ldots, N\}.$
Then $B_N(x,\varepsilon)$ is open with respect to the metric $d$.

### Problem 3

$S\in X$ is totally bounded in $(X,d)$ if and only if for any $\varepsilon > 0$ and any positive integer $N$ there exist $x_1, x_2, \ldots, x_m$ such that
$\displaystyle S \subset \bigcup_{l=1}^m B_N(x_l,\varepsilon)$.

### Problem 4

Let $V, U_j\subset \mathbb C$ open such that $V = \bigcup_{j=1}^\infty U_j$. Suppose $F:V\to\mathbb C$ is such that $F|_{U_j}$ is continuous in $U_j$. Then $F$ is continuous in $V$.

### Problem 5

Let $V\subset\mathbb C$ open and $S\subset C(V)$ pointwise bounded and pointwise equicontinuous. Then $S$ is uniformly bounded on any compact $K\subset V$.

### Comentarios

1. En el problema 4 no falta una hipótesis?

1. No, así está bien.

Nota que ya sabemos que F está definida en todo V.