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Cerrena [4.2K]
3 years ago
13

Knowing that a squence (a of n) is monotonically increasing and it's subsequence is convergent to a value A∈lR (for any n1 How d

o I prove that (a of n) converges to A aswell.
Mathematics
1 answer:
siniylev [52]3 years ago
8 0

Suppose {xn} is increasing and has a subsequence {xnk} which converges to L. We will prove that {xn} itself converges to L.

For any ϵ>0, we want to find an integer Nϵ such that |xn−L|≤ϵ for any n≥Nϵ.

Since {xnk} is increasing and converges to L, we can find kϵ such that for any k≥kϵ, −ϵ<xnk−L<0.

Take Nϵ=nkϵ, then for any n≥Nϵ, xnkϵ≤xn≤L, so −ϵ≤xnkϵ−L≤xn−L≤0.

Similarly, we can prove when {xn} is decreasing

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Hope it helps :)

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