Part 3/8:
To understand this definition better, one can visualize a number line where the limit ( L ) is marked. The interval around ( L ) defined by ( (L - \epsilon, L + \epsilon) ) represents the bounds within which the terms of the sequence must fall after a certain point. Regardless of how small ( \epsilon ) becomes, provided ( \epsilon > 0 ), a corresponding ( N ) exists allowing all terms ( a_n ) with ( n > N ) to reside within this interval.
For instance, plotting terms ( a_1, a_2, a_3, ) etc., on the number line evidences how they fluctuate but ultimately converge on the limit ( L ).