Analysis of theorems on sequence limits:
Let $\lim_{n \to \infty} a_n = A$ and $\lim_{n \to \infty} b_n = B$.

• Correct: By the fundamental limit theorem for sequences, if $(a_n)$ and $(b_n)$ converge, their sum $(a_n + b_n)$ converges and $\lim_{n \to \infty} (a_n + b_n) = A + B$.
• Incorrect: If $B \ne 0$, $(a_n / b_n)$ converges to $A / B$.
• Incorrect: If $a_n \to 0$, $(b_n / a_n)$ may diverge (e.g. division by zero limit).
• Incorrect: The difference $(a_n - b_n)$ is always convergent to $A - B$.

Mathematical evaluation of sequence limits:

• For sequence $(a_n) = n^2$:
  $$\lim_{n \to \infty} a_n = \lim_{n \to \infty} n^2 = +\infty$$
  Since the limit is not a finite real number, $(a_n)$ is divergent.
• For sequence $(b_n) = 1 + \frac{1}{n^2}$:
  $$\lim_{n \to \infty} b_n = \lim_{n \to \infty} \left(1 + \frac{1}{n^2}\right) = 1 + 0 = 1$$
  Since the limit exists as a finite real number ($1$), $(b_n)$ is convergent.
• Incorrect: $(a_n)$ does not converge.
• Incorrect: $(b_n)$ does not diverge.
• Incorrect: The roles are inverted.
• Correct: $(a_n)$ is divergent and $(b_n)$ is convergent.

Evaluation of sequence limit:

• Incorrect: The limit is 0, not 1.
• Correct: Taking the limit as $n \to \infty$:
  $$\lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{1}{n^2} = 0$$
  Because a finite unique limit exists ($L = 0$), the sequence $(a_n)$ is convergent and its limit is 0.
• Incorrect: The sequence is bounded and monotonically decreasing to 0, hence convergent.
• Incorrect: A convergent sequence in $\mathbb{R}$ possesses a unique limit, never multiple limits.