Real Analysis I am proving that the telescopic series (a 1 -a 2 )+(a 2 -a 3 XXXXXXXXXXconverges IFF the sequence {an} n=1 to infinity. I posted this question and got a response. I mostly understand it...

Real Analysis

I am proving that the telescopic series (a₁-a₂)+(a₂-a₃)+. . . converges IFF the sequence {an}_n=1to infinity.

I posted this question and got a response.  I mostly understand it but have a few questions.

I am told to have S_n=Sum  (a_i-a_i+1) from i=1 to n.  Do I call Sn the sequence of partial sums?   Because if the sequence of partial sums is convergent then the series converges.  If S_n
is not the sequence of partial sums, then what is it?  And do I say "Let S_n= Sum . . . "

In another part of the answer I was given,  when we suppose that the sequence {a_n} converges we arrive at

S_n=a₁-a_n+1

Then we take the limit of both sides.  I understand why the limit of the RHS exists but why does the limit of the LHS (S_n) exist?

Also, when I have proven  (2)sequence converges so series converges, how do I state the  conclusion? Do I say that since S_nis the sequence of partial sums of the series and Sn converges then the Sum (a_i-a_i+1) from i=1 to n converges and thus (a₁-a₂)+(a₂-a₃)+ .  . is convergent? Is that the appropriate wording?