If {x_n} is a convergent sequence in ℝ and {y_n} is a bounded sequence in ℝ, then we can conclude that

Concept:

(i) Every convergent sequence is bounded.

Explanation:

{xn} is a convergent sequence in ℝ. So it is bounded.

Then there exists a real number M such that |x_n| ≤ M.

 {yn} is a bounded sequence in ℝ

Then there exists a real number L such that |yn| ≤ L.

Now, |xn + yn| ≤ |xn| + |yn| ≤ M + L

So, {xn + yn} is bounded.

Option (2) is true.

Let {xn} = {\(\frac1n\)} and {yn} = {(-1)ⁿ} then {xn} is a convergent sequence in ℝ and {yn} is a bounded sequence in ℝ.

But {xn + yn} = {\(\frac1n\) + (-1)n} which is not convergent and it has convergent and bounded subsequence.

Options (1), (3) and (4) are false