exercise:0fce8ae2c1

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Let [math]s[/math] and [math]t[/math] be two infinite sequences and [math]a[/math] a real number such that

[[math]] s_n = a + t_n , [[/math]]

for every integer [math]n[/math] greater than or equal to some integer [math]k[/math]. Prove that

[[math]] \lim_{n\goesto\infty} s_n = a + \lim_{n\goesto\infty} t_n . [[/math]]

[Suggestion: It is easy to prove this result directly from the definition of convergence. Alternatively, one may consider a constant sequence with the single value [math]a[/math], and obtain the result as a corollary of Theorem (i).]