applicability: $\mathbb {N, Z, Q, R, C}$

# Proposition: Difference of Convergent Real Sequences

Let $$(a_n)_{n\in\mathbb N}$$ and $$(b_n)_{n\in\mathbb N}$$ be two convergent real sequences to the limits $$\lim_{n\rightarrow\infty} a_n=a$$ and $$\lim_{n\rightarrow\infty} b_n=b$$. Consider the real sequence $$(c_n)_{n\in\mathbb N}$$ with $$c_n:=a_n - b_n$$. Then $$(c_n)_{n\in\mathbb N}$$ is also convergent and its limit equals $$\lim_{n\rightarrow\infty} c_n=a - b$$.

### Notes

• This proposition can be expressed in the short form: $\lim_{n\rightarrow\infty} (a_n - b_n)=\lim_{n\rightarrow\infty} a_n - \lim_{n\rightarrow\infty} b_n.$
• The proposition's proof can be transferred also to sequences of other kinds than real numbers, for example, the complex numbers.

Proofs: 1

Proofs: 1 2

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### References

#### Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983