Primary objectives of sequence alignment and comparison in bioinformatics include:

• Identifying conserved motifs and shared functional patterns (Option 1).
• Inferring phylogenetic relationships and evolutionary ancestry (Option 3).
• Detecting evolutionary conservation across species (Option 4).
  
  Direct sequence comparison does not investigate the macroscopic physical characteristics (e.g., solubility, density, melting point) of molecules. Hence, Option 2 is not a purpose of sequence comparison.

Convergence of the sequence $(a_n) = \frac{1}{n^r}$:
If $r \gt 0$: $\lim_{n \to \infty} \frac{1}{n^r} = 0$, so $(a_n)$ converges to 0.
If $r = 0$: $a_n = \frac{1}{n^0} = 1$ for all $n$, so $(a_n)$ is a constant sequence converging to 1.
If $r \lt 0$: Let $r = -k$ where $k \gt 0$. Then $a_n = n^k \to \infty$ as $n \to \infty$, which diverges.

Fundamental theorems on real sequences:
Theorem: Every Cauchy sequence in $\mathbb{R}$ is bounded. (Option 4 is true).
Counter-example for Option 1 & 2: The sequence $a_n = (-1)^n$ is bounded ($-1 \le a_n \le 1$), but it is neither convergent nor a Cauchy sequence.
Option 3 is false: Because every Cauchy sequence is unconditionally bounded.