Limit question with Substituting

Fine the limit of (x^1000 - 1)/(x - 1) as x approaches 1.

Solution:


(x^1000 - 1)/(x - 1) = (x^999 + x^998 + ... + x + 1)(x - 1)/(x - 1)

= (x^999 + x^998 + ... + x + 1)

Substituting x = 1 I get....

Line (1):1 + 1₁ + 1₂ + ..... + 1₉₉₉ = 1 + 999 = 1000

Is there any way to prove Line (1): I know it is obvious but I want to prove it mathematically and with out obviously counting it on my fingers.

 

I guess I could say 1 + 1(999) = 1000

 

If any one has a better way of solving it please let me know, I want to learn all I can.