Statement: If ${a_n}$ and ${b_n}$ are two sequences of real numbers, then ${a_n + b_n}$ is a convergent sequence.
Counterexample: Let $a_n = 1$ for all $n$, $b_n = -1$ for all $n$. Both converge, but $a_n + b_n = 0$ for all $n$, which is convergent. However, the PDF says “option 1 is not correct,” which may refer to a different statement or a misinterpretation. The given explanation is not clear, but the PDF concludes: “Hence option 1 is not correct.”
Statement: If ${a_n}$ is an increasing sequence, then ${(-1)^n a_n}$ is a decreasing sequence.
Counterexample: $a_n = n$ for all $n$. Then ${(-1)^n a_n}$ is not decreasing.
Solution: Hence option 2 is not correct.
Statement: If ${a_n} \to a$, ${b_n} \to b$, and both $a, b$ are non-zero, then ${a_n b_n} \to ab$ must be non-zero.
Solution: This is correct.
Conclusion: Option 3 is correct.
Statement: If ${a_n} \to a$ and ${b_n} \to a$, then ${a_n - b_n} \to 0$.
Solution: This is correct.
Conclusion: Option 4 is correct.
Statement: If a sequence is divergent, then any subsequence is also divergent.
Counterexample: Let $a_n = n$ if $n$ is odd, $a_n = 1$ if $n$ is even. ${a_n}$ is divergent, but ${a_{2n}}$ is constant and hence convergent.
