正数数列 $u_1$、$u_2$、$u_3$、$\ldots$ 满足 $u_1>4$，且对于 $n\ge1$，

(a) 通过考虑 $u_{n+1}-4$，或用其他方法，用数学归纳法证明：对于所有正整数 $n$，均有 $u_n>4$。

(b) 证明当 $n\ge1$ 时，$u_{n+1}<u_n$。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

归纳假设 $u_k>4$ 后，把 $u_{k+1}-4$ 因式分解，逐一检查分子和分母的正负。

For $n=1$, $u_1>4$ is given.

Assume $u_k>4$ for some positive integer $k$. Then

All three factors $u_k-3$, $u_k-4$ and $2u_k$ are positive, so $u_{k+1}>4$.

The statement holds for $n=1$ and is inherited from $k$ to $k+1$. Therefore, by mathematical induction, $\boxed{u_n>4}$ for every positive integer $n$.

比较相邻两项的差，再使用(a)已经证明的严格下界。

For $n\ge1$,

since $u_n>4$ by part (a). Hence $\boxed{u_{n+1}<u_n}$ for $n\ge1$.