A sequence is a function on the naturals
A sequence of real numbers is nothing exotic: it is an ordered list a_1, a_2, a_3, … with one term for each natural number. Formally it is a function from the naturals to the reals, n -> a_n, and we usually write the whole thing as (a_n). The number n is the index (it tells you the position), and a_n is the term at that position. Order matters, and the list never ends.
Three running examples will follow us through this guide. The sequence a_n = 1/n marches 1, 1/2, 1/3, 1/4, … toward zero. The sequence b_n = (-1)^n bounces -1, 1, -1, 1, … forever. The sequence c_n = n climbs 1, 2, 3, … and runs off to infinity. Each behaves differently, and the whole point of the limit concept is to say precisely how.
“Eventually close” is the idea to capture
Intuitively, (a_n) converges to a number L if the terms get and stay arbitrarily close to L as n grows. For a_n = 1/n the terms get as near zero as you like and never drift back, so the limit is 0. The two words that carry the weight are arbitrarily (no fixed tolerance is enough — you must be able to beat every one) and stay (it is not enough to touch L once; the closeness must hold from some point on). This is the seed of convergence.
Picture a thin band of half-width epsilon drawn around L on the number line — an informal neighborhood of L. Convergence to L means: no matter how thin you make the band, all but finitely many terms fall inside it. For 1/n with L = 0 and band (-0.01, 0.01), every term from n = 101 onward is inside; shrink the band and you just need to start a little later.
How close is 1/n to 0? Want 1/n < 0.1 -> need n > 10 -> works from n = 11 Want 1/n < 0.01 -> need n > 100 -> works from n = 101 Want 1/n < 0.001 -> need n > 1000 -> works from n = 1001 Pattern: for ANY tolerance e > 0, take n > 1/e. Then 1/n < e, and it stays < e for every larger n too. The starting index moved, but it was always FINITE. That "for any e there is a starting index" is exactly convergence.
Why we need a definition at all
“Gets close” is suggestive but slippery. Does 1, 0, 1/2, 0, 1/3, 0, … converge to 0? It keeps returning to 0, yet it also keeps stepping away. Does b_n = (-1)^n converge? It is always near either -1 or 1, but never settles on one. Plain words cannot adjudicate these cases; we need a test sharp enough to give a yes or no every time. That sharpness is what mathematical analysis is built on.