Wesley Hunt wrh8639

Week #7 5.1.23

Problem

Prove that this subset of R is a subspace of R:

U₄ = the set of sequences that are eventually zero: there is an index N such that x_n = 0 for all n > N.

Solution

An example of such a sequence is (2, 1, 0, 0, 0,…), which makes it an element of U₄.

What we need to prove is that addition and scalar multiplication are closed under U₄, meaning that the sum or scalar product of any elements in U₄ is still in U₄.