Divergent Real Sequence to Positive Infinity/Examples/2^n/Proof 1
Jump to navigation
Jump to search
Example of Divergent Real Sequence to Positive Infinity
Let $\sequence {a_n}_{n \mathop \ge 1}$ be the real sequence defined as:
- $a_n = 2^n$
Then $\sequence {a_n}$ is divergent to $+\infty$.
Proof
Let $H \in \R_{>0}$ be given.
By Boundedness of Nth Powers, $\sequence {a_n}$ is unbounded above.
Hence:
- $\exists N \in \N: 2^N > H$
Then if $n > N$, we have:
- $2^n > 2^N > H$
and the result follows.
$\blacksquare$
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 4$: Convergent Sequences: Exercise $\S 4.29 \ (3) \ \text{(i)}$