Is it true that if $\{x_{n}\}_{n=1}^{\infty}$ is a finite union of Riesz sequences in a Hilbert space H, then $\{x_{n}\}$ itself will be a Riesz sequence? What about Frames and Bessel seuences, do we have the same result? (In case of Bessel sequences I beleive it is true).
Note: A sequence $\{x_{n}\}$ is Riesz sequence in H iff there exits $A,B>0$ such that for any scalar sequence $\{a_{n}\}$
$ A\sum_{n} |a_{n}|^2 \leq \|\sum_{n} a_{n}x_{n}\|^2 \leq B\sum_{n} |a_{n}|^2 $