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EDIT 1. Let $(x_n)$ be a sequence of rational numbers that converges to an irrational number $x$. Must it be the case that $x_1, x_2, \dots$ are all irrational?

2. Let $(y_n)$ be a sequence that converges to the rational number $y$. Define such a sequence where $y_1, y_2, \dots$ are all irrational.

now these are the 2 questions in my textbook i noticed when posting this that someone had asked the answer to 1) and a similar but less difficult version of 2.

a couple things i noted the defined answer for 1 he used $\pi$ and simply added more digits to it for each part of the sequence. so firstly id like to ask would that really be a full mark answer on a analysis final? secondly and more importantly i was wondering if there was a way to define this where the final number wasn't transcendental

for 1) i used $ (1+1/n)^n $ s.t $n \in \mathbb{N} $ for every n this sequence is rational but for $ n \to \infty $

$(1+1/n)^n = e $

My Question:(i) Is this true and if so is there a definable sequence that holds but doesnt define a transcendental number?

EDIT Golden ratio from Fibonacci sequence.

For 2) i picked $ 2^{1/n} $ s.t $ n>1$ and $n \in \mathbb{N}$ i believe every value of this sequence is irrational but the limit should be 1

My Question:(ii) does the above work? and if possible could i have anther example of a sequence where every value is irrational but the limit is rational?

EDIT: $\frac{\sqrt 2}{n}$

Sorry i didn't realize i missed the rationals in statement 1

1. Let $(x_n)$ be a sequence of rational numbers that converges to an irrational number $x$. Must it be the case that $x_1, x_2, \dots$ are all irrational?

2. Let $(y_n)$ be a sequence that converges to a rational number $y$. Define such a sequence where $y_1, y_2, \dots$ are all irrational.

Now these are the 2 questions in my textbook i noticed when posting this that someone had asked the answer to 1) and a similar but less difficult version of 2.

A couple things i noted the defined answer for 1 he used $\pi$ and simply added more digits to it for each part of the sequence. so firstly id like to ask would that really be a full mark answer on a analysis final? secondly and more importantly i was wondering if there was a way to define this where the final number wasn't transcendental.

For 1) I used $ (1+1/n)^n $ s.t. $n \in \mathbb{N} $. For every $n$ this sequence is rational but for $\lim_{ n \to \infty }(1+1/n)^n = e $.

My Question:  (i) Is this true and if so is there a definable sequence that holds but doesn't define a transcendental number?

EDIT: Golden ratio from Fibonacci sequence.

For 2) I picked $ 2^{1/n} $ s.t. $ n>1$ and $n \in \mathbb{N}$. I believe every value of this sequence is irrational but the limit should be 1.

My Question:  (ii) does the above work? and if possible could I have another example of a sequence where every value is irrational but the limit is rational?

EDIT: $\frac{\sqrt 2}{n}$.

EDIT 1. Let $(x_n)$ be a sequence of rational numbers that converges to an irrational number $x$. Must it be the case that $x_1, x_2, \dots$ are all irrational?

2. Let $(y_n)$ be a sequence that converges to the rational number $y$. Define such a sequence where $y_1, y_2, \dots$ are all irrational.

now these are the 2 questions in my textbook i noticed when posting this that someone had asked the answer to 1) and a similar but less difficult version of 2.

a couple things i noted the defined answer for 1 he used $\pi$ and simply added more digits to it for each part of the sequence. so firstly id like to ask would that really be a full mark answer on a analysis final? secondly and more importantly i was wondering if there was a way to define this where the final number wasn't transcendental

for 1) i used $ (1+1/n)^n $ s.t $n \in \mathbb{N} $ for every n this sequence is rational but for $ n \to \infty $

$(1+1/n)^n = e $

My Question:(i) Is this true and if so is there a definable sequence that holds but doesnt define a transcendental number?

For 2) i picked $ 2^{1/n} $ s.t $ n>1$ and $n \in \mathbb{N}$ i believe every value of this sequence is irrational but the limit should be 1

My Question:(ii) does the above work? and if possible could i have anther example of a sequence where every value is irrational but the limit is rational?

EDIT: $\frac{\sqrt 2}{n}$

Sorry i didn't realize i missed the rationals in statement 1

EDIT 1. Let $(x_n)$ be a sequence of rational numbers that converges to an irrational number $x$. Must it be the case that $x_1, x_2, \dots$ are all irrational?

2. Let $(y_n)$ be a sequence that converges to the rational number $y$. Define such a sequence where $y_1, y_2, \dots$ are all irrational.

now these are the 2 questions in my textbook i noticed when posting this that someone had asked the answer to 1) and a similar but less difficult version of 2.

a couple things i noted the defined answer for 1 he used $\pi$ and simply added more digits to it for each part of the sequence. so firstly id like to ask would that really be a full mark answer on a analysis final? secondly and more importantly i was wondering if there was a way to define this where the final number wasn't transcendental

for 1) i used $ (1+1/n)^n $ s.t $n \in \mathbb{N} $ for every n this sequence is rational but for $ n \to \infty $

$(1+1/n)^n = e $

My Question:(i) Is this true and if so is there a definable sequence that holds but doesnt define a transcendental number?

EDIT Golden ratio from Fibonacci sequence.

For 2) i picked $ 2^{1/n} $ s.t $ n>1$ and $n \in \mathbb{N}$ i believe every value of this sequence is irrational but the limit should be 1

My Question:(ii) does the above work? and if possible could i have anther example of a sequence where every value is irrational but the limit is rational?

EDIT: $\frac{\sqrt 2}{n}$

Sorry i didn't realize i missed the rationals in statement 1