Question

Show that \(X_n= e^{sin(5n)}\) has a convergent subsequence.
Please, help

Answer 1

@nerdguy2535

Answer 2

well from wolfram i get that the limit does not exist as n goes to infinity
but it does say \[\frac{1}{e} <e^{\sin(5n)}<e \]

Answer 3

which means we should be able to find a sub-sequence that converges to 1/e

Answer 4

or even e

Answer 5

i think

Answer 6

i don't know if that helps or not

Answer 7

http://www.wolframalpha.com/input/?i=y%3De%5E%28sin%285x%29%29
looks like the graph is bounded so the sequence is bounded
and there is a theorem you can use here called the bolzano-weiestrass theorem

Answer 8

Thank you very much.

Answer 9

i got it. since \(-1\leq sin(5x) \leq 1\)--> \(e^{sin(5x)}\) is bounded as: \(\dfrac{1}{e} \leq e^{sin(5x)}\leq e\) which show it is bounded --> it has a convergent subsequence.