limit superior and limit inferior point set topology iit jam 2018 real analysis gate Mathematics

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Real Analysis: Interior points, limit points, open sets, closed sets, bounded sets, connected sets, compact sets, completeness of R. Power series (of real variable), Taylor's series, radius and interval of convergence, term-wise differentiation and integration of power series.

limit superior and limit inferior point set topology iit jam 2018 real analysis gate Mathematics

tag 03082021
Question
11. Let a_n {█( 2+(-1)^((n-1)/2)/n , if n is odd @ 1+1/2^n , if n is even ), n ∈ N.┤
Then which one of the following is TRUE?
(a) sup⁡{a_n |n∈N }=3 and inf⁡{a_n | n∈N}=1 (b) lim⁡inf⁡(a_n ) =lim sup⁡(a_n )=3/2
(c) sup⁡{a_n |n∈N}=2 and inf⁡{a_n | n∈ N}=1 (d) lim⁡inf⁡(a_n ) =1 and lim⁡sup⁡〖(a_n )=3〗

Concept Used:

(1) Limit Superior = largest limit point when Sequence is bounded
(2) Limit Inerior = Smallest limit Point when Sequence is bounded

En Jam MA 2018, 11
Key (a)