Conjectures and squeeze theorem Consider the following list of conjectures. provide a short proof for those that are true and a counter example for any that are false.
(a) If  $\lim(a_{n}-b_{n})=0$, then $\lim a_{n}=\lim b_{n}$.
(b) If $(b_{n})\rightarrow b$, then $|b_{n}|\rightarrow |b|$.
(c) If $(a_n) \rightarrow 0$ and $(b_n -a_n) \rightarrow 0$, then $(b_n) \rightarrow a$.
(d) If $(a_n) \rightarrow 0$ and $|b_n - b| \leq a_n$ for all $n \in N$, then $(b_n) \rightarrow b$
My solution attempt:
(a) False. Consider two sequence $(a_n)$ and $(b_n)$ with $a_n = b_n = (−1)^n$
.
(b) True. Suppose limn→∞ bn = b. Let ε > 0. Then there exists
N ∈ N such that n ≥ N implies
|bn − b| ≤ ε.
But we have
||bn| − |b|| ≤ |bn − b|
for all n and in particular for every n ≥ N. Thus
||bn| − |b|| ≤ ε
for every n ≥ N. Therefore, limn→∞ |bn| = |b|.
(c) True. (“(an) → 0” should be “(an) → a” in this problem.) If
(an) → a and (bn − an) → 0, then by the algebraic limit theorem,
lim bn = lim[an + (bn − an)] = lim an + lim bn − an = a + 0 = a
(d)
I know that the squeeze theorem is applicable here but I am unsure how to go about it. If someone could demonstrate how to use it appropriately, that would help.

For (d), 
$$0\leq |b_n-b| \leq a_n$$
and since $a_n\to 0$, the squeeze theorem implies that the middle term tends to zero as well, so $b_n\to b$.
