Convergent Sequences in Complex Unitary Space

For simplicity, we adopt the following convention: X is a complex unitary space, x, y, w, g, g1, g2 are points of X, z is a Complex, q, r, M are real numbers, s1, s2, s3, s4 are sequences of X, k, n, m are natural numbers, and N1 is an increasing sequence of naturals. Let us consider X, s1. We say that s1 is convergent if and only if: (Def. 1) There exists g such that for every r such that r > 0 there exists m such that for every n such that n ­ m holds ρ(s1(n), g) < r. Next we state several propositions: (1) If s1 is constant, then s1 is convergent. (2) If s2 is convergent and there exists k such that for every n such that k ¬ n holds s3(n) = s2(n), then s3 is convergent. (3) If s2 is convergent and s3 is convergent, then s2 + s3 is convergent. (4) If s2 is convergent and s3 is convergent, then s2 − s3 is convergent. (5) If s1 is convergent, then z · s1 is convergent.