Proposition: Sum of a Convergent Real Sequence and a Real Sequence Tending to Infininty

Let $(a_n)_{n\in\mathbb N}$ be a real sequence tending to infinity (i.e. either $+\infty$ or $-\infty$ ). Let $(b_n)_{n\in\mathbb N}$ be a real sequence tending to some real number $b$ , i.e. with $\lim_{n\to\infty} b_n=b.$ Then the real sequence $(a_n+b_n)_{n\in\mathbb N}$ is tending to infinity as $(a_n)_{n\in\mathbb N}$ does.

Proofs: 1

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