Problem involving limsup.

Question

I had a problem while computing the limit superior below:

If $r\sqrt[\leftroot{-2}\uproot{2}n]{|a_n|}\leq \sqrt[\leftroot{-2}\uproot{2}n]{M}$ $~~~~~~\forall n$
letting $n\rightarrow \infty$ $\implies$ $limsup$ $r\sqrt[\leftroot{-2}\uproot{2}n]{|a_n|}\leq 1$

But I can't get how is limsup$ \leq 1$ . Kindly help.

Answer 1

You can get that conclusion because $M$ is a fixed number.

$$limsup~ r \sqrt[n]{a_n} \le limsup ~ \sqrt[n]{M}= lim ~ \sqrt[n]{M} = 1$$