Result of Bounded Sequence in Equivalent Metrics | L7 | TYBSc Maths | Completeness ‪@ranjankhatu‬

hi everyone in this video we are going to discuss this example okay so what we have we have two Matrix D1 and D2 both are defined on same space X they are given these two Matrix are equivalent as well as they satisfy this inequality also right K1 and K2 are positive real numbers and for that we get this inequality and we have to prove that these two statements are equivalent sequence x n is bounded in X D1 if and only if same sequence is bounded in X D2 so what will I do first of all I will write a given information we have this is a very important information so I am writing it first so K 1 d 1 of x y less than or equal to D2 of x y less than or equal to K2 D1 of x y and this is true for all X okay so this is star this is so much important information for us ah K 1 and K 2 are positive real numbers so that is also given thing but I am not mentioning here so we will keep in mind K1 and K2 are positive we are going to use it later now we have to prove that these two statements are equivalent x n is bounded in X D1 if and only if x n is bounded in x d two so when we have if and only if condition what we do first we assume the first part that means I will assume x n is bounded in X to D1 we will prove x n is bounded in x d 2 and after that we will prove its Converse part so let us start assume that assume that what I am assuming x n is bounded in x d 1 and now our task is to prove that sequence x n is bounded in X D2 okay so it is bounded sequence what is definition of bounded sequence when we say the sequence is bounded if