Determine whether the following statement are true or false

Determine whether the following statement are true or false and show why.

Solution

a) It is given that {xn} and {yn} is strictly increasing then {xn +yn} will also increase so the statement is true.

For ex- If the sequence {x_n}=a² and the sequence {y_n}=a³ then the sequence {x_n + y_n} will be

{a²+a³} which is an increasing sequence

b) Similarly if {x_n} and {y_n} is strictly increasing then {x_n x y_n} will also increase

using above example we can see if {x_n}=a² and {y_n}=a³ then

{x_n x y_n} will be a² x a³= a⁵ which is also strictly increasing hence statement second is also true.

c) First we need to know what is a monotonic sequence :-

  A sequence (x_n) in R is increasing, if x_n x_n+1 for all n N, and decreasing if x_n+1 x_n for all n N. If (x_n) is either increasing or decreasing, then it is called monotone.

So we see that in the given question {x_n} and {y_n} both are monotonic then according to the definition we can see that {x_n + y_n} will be a monotonic sequence.