If two sequences $\{a_n\}$ and $\{b_n\}$ be such that $a_n>b_n$ then show that $\lim a_n\ge\lim b_n$

$$ \text{Take two converging sequences } \{a_n\}_{n=1}^\infty \text{ and } \{b_n\}_{n=1}^\infty, \text{ such that } a_n > b_n \text{ for all } n. \text{ Show that } \lim_{n \to \infty} a_n \geq \lim_{n \to \infty} b_n. \text{ Can we replace the weak inequality with strict inequality?} $$

Here is my approach to this proof...and I have the following questions

1. Is my proof correct?
2. Is there a better way to prove this?

My proof

Let $$\{a_n\} \rightarrow a \quad and \quad \{b_n\} \rightarrow b$$

Then by definition of convergence we have,

$$ \forall \epsilon > 0 \; \exists \;N_a \; and \; N_b \in \mathbb{N} \;such \;that $$

\begin{equation} a-\epsilon < a_n < a+\epsilon \quad \forall n\ge N_a \tag{1}\end{equation}

\begin{equation}b-\epsilon < b_n < b+\epsilon \quad \forall n\ge N_b \tag{2}\end{equation}

Now, let us assume that $$b>a$$which implies $$b - \epsilon > a - \epsilon \quad and \quad b + \epsilon > a + \epsilon \tag{3} $$

We also have that, $$a_n > b_b$$

Now from (1) and (2) we can say that

$$a_n > b_n \quad \text{and} \quad b_n > b - \epsilon \\\Rightarrow a_n > b - \epsilon \\ \text{and}\\a_n < a + \epsilon < b + \epsilon \quad [\text{from (3)}] \\\text{Therefore, we have for all}\quad \epsilon \ge 0 \quad \exists N = \max\{N_a,N_b\} \quad\text{such that}\\ b - \epsilon < a_n < b + \epsilon \quad\forall n \ge N \\ \Rightarrow \quad \{a_n\} \rightarrow b \\ \Rightarrow a = b \quad \text{[Since, two distinct sequences can't converge to more than one point.]}$$

But, this contradicts our assumption that b > a, so our supposition was wrong and b must be less than or equal to a.

Moreover we cant repla]ce the weak inequality with strict inequality because we can come up with a sequences $$\{a_n\} \rightarrow a \quad \text{and}\quad \{b_n\}\rightarrow b$$ such that $$a = b$$

One such example is$$ \{a_n\} = 1 + \frac{1}{n} \rightarrow 1 \\\{b_n\} = 1 + \frac{1}{n+1} \rightarrow 1 \\\text{and}\quad a_n > b_n \forall n \in \mathbb{N}$$

Also, this is my first time asking a question in math stack exchange so any tip/advice on the way I've presented my question is welcome.Thank you.