Prove or disprove:

If http://latex.codecogs.com/png.latex?(x_ny_n) is a null sequence then one or both of http://latex.codecogs.com/png.latex?(x_n), \, (y_n) is(are) null.

My attempt:

if http://latex.codecogs.com/png.latex?..., and \, (y_n) both are null then certainly http://latex.codecogs.com/png.latex?(x_ny_n) is null, no problem with that.

I assumed that http://latex.codecogs.com/png.latex?(x_n) is not null, http://latex.codecogs.com/png.latex?(y_n) is not null and as given, http://latex.codecogs.com/png.latex?(x_ny_n) is null. I tried to bring a contradiction since i have a feeling that the statement in the question is true.

since its assumed that http://latex.codecogs.com/png.latex?(x_n) is not null so:

there existshttp://latex.codecogs.com/png.latex?\epsilon_1>0such that for allhttp://latex.codecogs.com/png.latex?Nthere existshttp://latex.codecogs.com/png.latex?n \geq Nsuch thathttp://latex.codecogs.com/png.latex?|x_n|>\epsilon_1

similarly:

there existshttp://latex.codecogs.com/png.latex?\epsilon_2>0such that for allhttp://latex.codecogs.com/png.latex?Nthere existshttp://latex.codecogs.com/png.latex?n \geq Nsuch thathttp://latex.codecogs.com/png.latex?|y_n|>\epsilon_2

and since http://latex.codecogs.com/png.latex?(x_ny_n) is null:

there existshttp://latex.codecogs.com/png.latex?Nsuch thathttp://latex.codecogs.com/png.latex?...n_1 \epsilon_2

now since i can't say that http://latex.codecogs.com/png.latex?..._n|>\epsilon_2 occur at thesamehttp://latex.codecogs.com/png.latex?n i can't find a contradiction.

someone help.