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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