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#### issacnewton

##### Member

- Jan 30, 2012

- 61

I want to prove the following.

Let \(X\) and \(Y\) be two sequences,and \(XY\) converges. Then prove that

\(X_mY\) also converges,where

\[ X_m = \mbox{ m-tail of X } = (x_{m+n}\;:\; n\in \mathbb{N}) \]

Here is my proof.

let \(\lim\;(XY) = a \) . Then we have

\[ \forall \varepsilon >0\; \exists K_1 \in \mathbb{N} \;\forall n\geqslant K_1 \]

\[ |x_ny_n - a| < \varepsilon \]

Now let \(K = \max(K_1,\; m+1) \). The \( \forall n \geqslant K \) we have

\[|x_n y_n - a| < \varepsilon \].

But now all the \( (x_n) \) terms are values from the sequence \(X_m\). So

we have proven that

\[ \forall \varepsilon >0 \;\exists K \in \mathbb{N} \;\forall n\geqslant K \]

\[ |x_ny_n - a| < \varepsilon \]

where \( x_n\) values are from sequence \(X_m\). This proves that

\[ \lim(X_m Y) = a \]

which proves that \(X_mY\) also converges. Let me know if this is right

Thanks