Final answer:
The question pertains to a vector space in Mathematics, specifically sequences with an infinite tail of zeroes within the space of real numbers.
Step-by-step explanation:
The student is asking about a concept in Mathematics, specifically about a vector space over the field of real numbers (denoted by \( \mathbb{R} \)). The vector space in question consists of all sequences that eventually become zero and remain zero thereafter (referred to as having an 'infinite tail of zeroes').
Formally, a sequence \( (a_n) \) belongs to this space if there exists a non-negative integer \( N \) such that for all \( n > N \), the sequence elements \( a_n = 0 \). This property ensures that beyond a certain point, the sequence contains only zeroes. Examples of sequences in this space include \( (1, 2, 0, 0, 0, \ldots) \) and \( (3, -1, 4, 0, 0, 0, \ldots) \), where after a finite number of terms, the sequence is entirely composed of zeroes.