Csir net june 2018: Real analysis Question (1)

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General Mathematics
Question:

Given $(x_n)$ is convergent sequence in $\mathbb{R}$ and $(y_n)$ is bounded sequence in $\mathbb{R}$ then we can conclude that,

(a) $(x_n+y_n)$ is convergent.
(b) $(x_n+y_n)$ is bounded.
(c) $(x_n+y_n)$ has no convergent subsequence.
(d)$(x_n+y_n)$ has no bounded subsequence.

Solution: we discuss options one by one

(a) take $x_n=1$ for all $n\in\mathbb{N}$ and $y_n=(-1)^n$ for all $n\in\mathbb{N}$ then, sequence $(x_n)$ is convergent and $(y_n)$ is bounded. But, $(x_n+y_n)=(0,2,0,2,0,2,....)$ which is not convergent. Hence (a) is false.

(b) given that sequence $(x_n)$ is convergent and hence it is bounded. Also given that, $(y_n)$ is bounded. So that sum of two bounded sequence $(x_n+y_n)$ is also bounded. Hence (b) is true.

(c), (d): By option (b) as sequence $(x_n+y_n)$ is bounded sequence and hence by "Bolzano Weierstrass theorem for sequences", it has convergent (hence bounded) subsequence. 
So that, (c),(d) are false. 

Hence answer is option (b) only. 













 

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