MHB Sequence has convergent subsequence

[evinda]

Hello! (Wave)

Let $(a_n), (b_n), (c_n)$ sequences such that $(a_n), (c_n)$ are bounded and $a_n \leq b_n \leq c_n$ for each $n=1,2, \dots$ I want to show that $(b_n)$ has a convergent subsequence.

I have thought the following:

Since $(a_n), (c_n)$ are bounded, $\exists m_1, m_2 \in \mathbb{Z}$ such that $m_1 \leq a_n$ and $c_n \leq m_2$.

Then $m_1 \leq b_n \leq m_2$, i.e. $(b_n)$ is bounded. So, from Bolzano-Weierstrass theorem, $(b_n)$ has a convergent subsequence.

But... is the above complete? Could we also show it somehow else? (Thinking)

 

[GJA]

Hey evinda,

Looks good to me.

 

[evinda]

Hey evinda,

Looks good to me.

Nice, thank you... (Smirk)