Please write your answers in a legible way and submit a single pdf file. 1. Consider functions $$ f_{1}, f_{2}: \mathbb{N} \rightarrow \mathbb{R} $$ which are asymptotically positive. Using the definitions show that if $$ f_{1}=O\left(f_{2}\right) $$ then $$ f_{1}+f_{2}=\Theta\left(f_{2}\right) $$.

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**Title:** Proof that $$ f_{1} + f_{2} = \Theta\left(f_{2}\right) $$ Given $$ f_{1} = O\left(f_{2}\right) $$

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**Introduction**

In the analysis of algorithms and asymptotic behavior of functions, understanding the relationship between different growth rates is crucial. This proof demonstrates that for two asymptotically positive functions $$ f_{1}, f_{2}: \mathbb{N} \rightarrow \mathbb{R} $$, if $$ f_{1} = O\left(f_{2}\right) $$, then the sum $$ f_{1} + f_{2} $$ is $$ \Theta\left(f_{2}\right) $$.

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**Definitions**

1. **Big-O Notation ($$ O $$)**:
   - **Formal Definition**: $$ f_{1} = O\left(f_{2}\right) $$ if there exist positive constants $$ c $$ and $$ n_{0} $$ such that for all $$ n \geq n_{0} $$,
     $$
     f_{1}(n) \leq c \cdot f_{2}(n)
     $$
   
2. **Big-Theta Notation ($$ \Theta $$)**:
   - **Formal Definition**: $$ f = \Theta\left(g\right) $$ if and only if $$ f = O\left(g\right) $$ and $$ f = \Omega\left(g\right) $$. Formally, there exist positive constants $$ c_{1}, c_{2} $$ and $$ n_{0} $$ such that for all $$ n \geq n_{0} $$,
     $$
     c_{1} \cdot g(n) \leq f(n) \leq c_{2} \cdot g(n)
     $$

3. **Little-O Notation ($$ o $$)** (for completeness):
   - **Formal Definition**: $$ f = o\left(g\right) $$ if for every positive constant $$ c $$, there exists a positive integer $$ n_{0} $$ such that for all $$ n \geq n_{0} $$,
     $$
     0 \leq f(n) < c \cdot g(n)
     $$

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**Proof**

**Given:** 
- $$ f_{1}, f_{2}: \mathbb{N} \rightarrow \mathbb{R} $$ are asymptotically positive functions.
- $$ f_{1} = O\left(f_{2}\right) $$.

**To Prove:** 
- $$ f_{1} + f_{2} = \Theta\left(f_{2}\right) $$.

**Proof Steps:**

1. **Upper Bound ($$ f_{1} + f_{2} = O\left(f_{2}\right) $$):**

Since $$ f_{1} = O\left(f_{2}\right) $$, by definition, there exist constants $$ c > 0 $$ and $$ n_{0} $$ such that for all $$ n \geq n_{0} $$,
   $$
   f_{1}(n) \leq c \cdot f_{2}(n)
   $$
   
   Adding $$ f_{2}(n) $$ to both sides:
   $$
   f_{1}(n) + f_{2}(n) \leq c \cdot f_{2}(n) + f_{2}(n) = (c + 1) \cdot f_{2}(n)
   $$
   
   Therefore, 
   $$
   f_{1} + f_{2} \leq (c + 1) \cdot f_{2}
   $$
   
   This satisfies the definition of $$ f_{1} + f_{2} = O\left(f_{2}\right) $$ with constants $$ c' = c + 1 $$ and the same $$ n_{0} $$.

2. **Lower Bound ($$ f_{1} + f_{2} = \Omega\left(f_{2}\right) $$):**

Since both $$ f_{1} $$ and $$ f_{2} $$ are asymptotically positive, for all $$ n $$ sufficiently large, $$ f_{2}(n) > 0 $$.

Consider the sum:
   $$
   f_{1}(n) + f_{2}(n) \geq f_{2}(n)
   $$
   
   This inequality holds because $$ f_{1}(n) \geq 0 $$ (as $$ f_{1} $$ is asymptotically positive).
   
   Thus, 
   $$
   f_{1} + f_{2} \geq 1 \cdot f_{2}(n)
   $$
   
   This satisfies the definition of $$ f_{1} + f_{2} = \Omega\left(f_{2}\right) $$ with constant $$ c_{1} = 1 $$ and the same $$ n_{0} $$.

3. **Combining Upper and Lower Bounds:**

From steps 1 and 2, we have:
   $$
   1 \cdot f_{2}(n) \leq f_{1}(n) + f_{2}(n) \leq (c + 1) \cdot f_{2}(n)
   $$
   
   Therefore, by definition:
   $$
   f_{1} + f_{2} = \Theta\left(f_{2}\right)
   $$

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**Conclusion**

By demonstrating both the upper and lower bounds, we have shown that if $$ f_{1} = O\left(f_{2}\right) $$ and both functions are asymptotically positive, then the sum $$ f_{1} + f_{2} $$ grows asymptotically at the same rate as $$ f_{2} $$. Hence, $$ f_{1} + f_{2} = \Theta\left(f_{2}\right) $$.
If $$ f_{1} = O(f_{2}) $$ and both functions are asymptotically positive, then $$ f_{1} + f_{2} = \Theta(f_{2}) $$.

Please write your answers in a legible way and submit a single pdf file. 1. Consider functions \( f_{1}, f_{2}: \mathbb{N} \rightarrow \mathbb{R} \) which are asymptotically positive. Using the definitions show that if \( f_{1}=O\left(f_{2}\right) \) then \( f_{1}+f_{2}=\Theta\left(f_{2}\right) \).

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