Question

All subsequences of a convergent sequence of real numbers convergent to the same limit​

Answer 1

Answer:

Theorem 3.4 If a sequence converges then all subsequences converge and all convergent subsequences converge to the same limit. Proof Let {an}n∈N be any convergent sequence. ... Let {bn}n∈N be any subsequence

Answer 2

Let $s_{n_k}$ denote a subsequence of $s_n$.

Note that $n_k \geqslant k$

for all k. This easy to prove by induction: in fact, $n_1 \geqslant 1$ and $n_k \geqslant k$ implies $n_{k + 1} > n_k \geqslant k$ and hence $n_{k + 1} \geqslant k + 1$

Let $lim \: s_n = s$

and let $\epsilon > 0$

There exists N so that

$n > N$ implies $|s_n - s| < N \epsilon$.

Now $k > N$

⟹ $n_k > N$

⟹$|s_{n_k} - s| < \epsilon$

Therefore: $lim \: s_{n_k} = s$

Where k tends to ∞.

Therefore,all subsequences of a convergent sequence of real numbers convergent to the same limit.

This is a problem of Algebra.

Some important Algebra formulas.

(a + b)² = a² + 2ab + b²

(a − b)² = a² − 2ab − b²

(a + b)³ = a³ + 3a²b + 3ab² + b³

(a - b)³ = a³ - 3a²b + 3ab² - b³

a³ + b³ = (a + b)³ − 3ab(a + b)

a³ - b³ = (a -b)³ + 3ab(a - b)

a² − b² = (a + b)(a − b)

a² + b² = (a + b)² − 2ab

a² + b² = (a − b)² + 2ab

a³ − b³ = (a − b)(a² + ab + b²)

a³ + b³ = (a + b)(a² − ab + b²)

Know more about Algebra,

1) https://brainly.in/question/13024124

2) https://brainly.in/question/1169549