A model describing certain entities is formed by associating with an entity a sequence of values (possibly infinite), each term in the sequence being associated with a particular characteristic of the entity. For example - for a sequence s describing an entity (a person, say), s₀ might stand for how right-wing in their political views that person is. For example if s₀ = 5 this might represent the fact that the person is moderately right-wing while if s₀ = -5 then we might interpret this as meaning the person is moderately lest-wing. The model wishes to allow for expansion to include extra characteristics as time passed and hence it is reasonable that the characteristic sequence be modelled as infinite sequences. In comparing entities we need to have a metric that allows us determine how similar or dissimilar entities are. The following is a popular metric d(s, t) = ?|s_n-t_n|/sⁿ(1+|s_n-t_n|) where s and t are sequences (describing entities) and the summation is over the range 0........?.

Show that this measure (of similarity) d satisfies the triangle inequality, namely that, d(s, t)

Question 2: A model describing certain entities is formed by asso- ciating with an entity a sequence of values (possibly innite), each term in the sequence being associated with a particular characteris- tic of the entity. For example - for a sequence .9 describing an entity (a person, say), so might stand for how right-wing in their political views that person is. For example if so = 5 this might reprsent the fact that the perosn is moderately right-wing while if so = 5 then we might interpret this as meaning the person is moderately left- wing. The model wishes to allow for expansion to include extra characteristics as time passes and hence it is reasonable that the characterisitic sequence be modelled as innite sequences. In com- paring entities we need to have a metric that allows us determine how similar or dissimilar entities are. The following is a popular metric wheer s and t are sequences (descrribing entities) and the sum- mation is over the range 0 ...... oo. . Show that this measure (of similarity) d satises the Inequality, namely that, d(s,t) 5 d(s,u) + d(u,t) this the set of sequences is called a metric space - and its elements are the sequences that encode the characteristics of the entities. ) triangle . (Note: Because of