1. ## Help with Proofs

I'm taking a new class and instructor isn't very helpful . These are probably very simple, but I'm lost.

1) If an-->L, then |an|-->|L|

2) Prove that lim(an-bn) = lim(an)-lim(bn), provided lim(an) and lim(bn) exist

3) Give an example where lim(an) and lim(bn) do not exist, but lim(an+bn) exists.

4) If {an} is a bounded sequence and if {bn} is a sequence converging to 0, then {anbn} converges to 0.

Help is extremely appreciated!

I'm taking a new class and instructor isn't very helpful . These are probably very simple, but I'm lost.

1) If an-->L, then |an|-->|L|

2) Prove that lim(an-bn) = lim(an)-lim(bn), provided lim(an) and lim(bn) exist

3) Give an example where lim(an) and lim(bn) do not exist, but lim(an+bn) exists.

4) If {an} is a bounded sequence and if {bn} is a sequence converging to 0, then {anbn} converges to 0.
The secret to #1 is $\left| {\left| {a_n } \right| - \left| L \right|} \right| \leqslant \left| {a_n - L} \right|$.

For #4 Suppose that $\left( {\forall n} \right)\left[ {\left| {a_n } \right| \leqslant A} \right]$ then $\left| {a_n b_n } \right| = \left| {a_n } \right|\left| {b_n } \right| \leqslant A\left| {b_n } \right|$

I'm taking a new class and instructor isn't very helpful . These are probably very simple, but I'm lost.

1) If an-->L, then |an|-->|L|

2) Prove that lim(an-bn) = lim(an)-lim(bn), provided lim(an) and lim(bn) exist

3) Give an example where lim(an) and lim(bn) do not exist, but lim(an+bn) exists.

4) If {an} is a bounded sequence and if {bn} is a sequence converging to 0, then {anbn} converges to 0.

2. Since $\{a_n\}\to a$, $\exists~N_1$ such that $n>N_1$ implies that $|a_n-a|<\frac{\epsilon}{2}$. Similarly, since $\{b_n\} \to b$, $\exists~N_2$ such that $n>N_2$ implies that $|b_n-b|<\frac{\epsilon}{2}$.
Let $N=\max\{N_1,N_2\}$. Thus, when $n>N, |(a_n-b_n)-(a-b)| = \underbrace{|(a_n-a)+(b-b_n)| \leq |a_n-a|+|b-b_n|}_{triangle~inequality} < \frac{\epsilon}{2}+\frac{\epsilon}{2}=\epsilon$.
Therefore $\lim(a_n-b_n)=a-b = \lim a_n-\lim b_n$.
3. Consider $a_n = 1+(-1)^n$ and $b_n=1-(-1)^n$.