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## Main Question or Discussion Point

Suppose I have a separable Hilbert space [itex]\mathcal H[/itex] and vectors [itex]x_1(p),x_2(p),y_1(p),y_2(p) \in \mathcal H[/itex] that depend on a parameter [itex]p>0[/itex] such that

[tex]

\| x_1 - y_1 \| \to 0 \qquad \text{as $p \to 0$}

[/tex]

and

[tex]

\| x_2 - y_2 \| \to 0 \qquad \text{as $p \to 0$}.

[/tex]

Can anything be said about [itex]|(x_1,x_2) - (y_1,y_2)|[/itex]? I'd like to be able to say it goes to zero as [itex]p\to 0[/itex], but I haven't been able to show that yet...

[tex]

\| x_1 - y_1 \| \to 0 \qquad \text{as $p \to 0$}

[/tex]

and

[tex]

\| x_2 - y_2 \| \to 0 \qquad \text{as $p \to 0$}.

[/tex]

Can anything be said about [itex]|(x_1,x_2) - (y_1,y_2)|[/itex]? I'd like to be able to say it goes to zero as [itex]p\to 0[/itex], but I haven't been able to show that yet...