1.(2 points) Let $a_{nm}$ be a double sequence of real numbers, such that $\lim_{m \to \infty} a_{nm} = 1$ and $\lim_{n \to \infty} a_{nm} = 0$. Is it true that $\lim_{n \to \infty} a_{nn} $ exists and is in $[0, 1]$? If you think this is true, prove it. Otherwise find a counter example.

Let $f: X \to \R$ be a function on a metric space. We say $f$ is Lipschitz continuous, if there exists a $K>0$, such that for any $x, y \in X$, we have $$ |f(x) - f(y) | \leq K d(x,y). $$ Such a $K$ is called 'an Lipschitz constant' for $f$.

2. (2 points) Prove that if $f$ is Lipschitz continuous then $f$ is uniformly continuous.

3. (2 points) Let $f_n: X \to \R$ be a sequence of functions on a metric space $X$. Suppose $f_n$ converges to $f$ pointwise, and $f_n$ are Lipschitz continuous with a common $K$ as a Lipschitz constant, namely, for any $x, y \in X$, and any $n \in \N$, we have $ |f_n(x) - f_n(y) | \leq K d(x,y). $ Is it true that $f_n$ converges to $f$ uniformly? If true, prove it; otherwise find counterexample.

4. (2 points). Prove that $f_n(x) = \frac{\sin (x)}{1 + n x^2}$ converges uniformly on $\R$.

5. (2 points). Let $f_n, g_n: X \to \R$ be continuous functions. If $f_n \to f$ and $g_n \to g$ converges uniformly, is it true that $f_n g_n \to fg$ uniformly? If true, prove it; otherwise find counterexample.