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enter image description here In the first question option ( a ) is incorrect as given in the answer . can anyone give a counter example to substantiate it. And in the second question I really don't agree with option (b) as if a_n = 1/n ^2 here it converges but √(1/n^2) = 1/n is divergent . so it must be false. Isn't it?. Correct me if I am wrong.

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    Reasoning for $(b)$ is good. – 2017-01-22

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For the first part of your question take two sequences that are "zippered" if you will. Example $a_n= \frac{1}{n}$ for odd terms, and $1$ for the evens, and $b_n= \frac{1}{n}$ for even terms, and $1$ for the odds. Then neither of these converges on their own, but their sums and products do. As for part $b$, your reasoning is correct.