Let X, Y be defined by

​$$X = \{(x_n)_{n \geq 1} : \limsup_{n \to \infty} x_n = 1, \; \text{where} \; x_n \in \{0, 1\}\}$$​​

and

​$$Y = \{(x_n)_{n \geq 1} : \lim_{n \to \infty} x_n \; \text{does not exist, where} \; x_n \in \{0, 1\}\}$$​.

Which of the following is true?

Correct option is D

Let $$x_n:\mathbb{N}\to \{0,1\}$$​  be a function(sequence).
total no. of such functions=$$2^{\textit{cardinality of natural numbers}}$$​  = uncountable
now if

​$$\limsup_{n\to \infty}(x_n)=1$$​

then  $$X = \{(x_n)_{n \geq 1} : \limsup_{n \to \infty} x_n = 1, \; x_n \in \{0, 1\}\}$$​  is an uncountable set as after any finite terms and up to infinite terms, $$x_n$$​ can take both values 0 and 1. 

Similarly, 

​$$Y = \{(x_n)_{n \geq 1} : \lim_{n \to \infty} x_n \; \text{does not exist, where} \; x_n \in \{0, 1\}\}$$​​

is an uncountable set as both 0 and 1 are possible values of $$x_n$$​ up to infinite terms.

​$$\textbf{Correct Answer: (D).}$$​​