text: Q:
  
  "The limit of a sequence is insensitive to finite changes in the sequence" - help me understand this sentence!
  
  The context is p59 of "A primer on Hilbert Space Theory, by Alabiso and Weiss, where the example is given of a quotient $c_{00}/c_0$ which is 
  "the quotient of the space $c_0$, the space of all converging sequences of (say) real numbers, by the subspace of all eventually $0$ sequences [i.e. all sequences like $(a_1, a_2, ..., a_i, 0, 0, 0, 0, ...)$]. The quotient may be thought of as the space of all sequences modulo finite changes. In analysis it is well known that the limit of a sequence is insensitive to finite changes in the sequence, and so it is precisely this quotient that is of interest already in elementary analysis."
  The text italics is what I'm having trouble with - what kind of 'finite changes' are they talking about that would not change the limit of the sequence? Sorry if this is something really obvious, I'm new to this area and it sort of defies my intuition.
  
  A:
  
  Suppose your sequence converges to a limit $L$; then given any small $\epsilon >0$ you know after a fixed number $N\in\mathbb{N}$ all terms $a_n$ are less than a distance of $\epsilon$ from $L$, when $n\geq N$.
  By changing a finite number of elements of the sequence, all you do to this definition of limit is possibly increase the size of the threshold value $N$. However, because you only changed finitely many terms this threshold still exists and since you can always find infinitely many natural numbers $n\geq N$ your limit $L$ is unchanged. 
  
  